Estimating the Gerber-Shiu Function in a Compound …downloads.hindawi.com/journals/ddns/2019/5071268.pdfGerber-Shiu function is proposed by the observed sample of the surplus process

Research ArticleEstimating the Gerber-Shiu Function in a Compound PoissonRisk Model with Stochastic Premium Income

YunyunWang1 Wenguang Yu 2 and YujuanHuang 3

1College of Mathematics and Statistics Chongqing University Chongqing 401331 China2School of Insurance Shandong University of Finance and Economics Jinan 250014 China3School of Science Shandong Jiaotong University Jinan Shandong 250357 China

Correspondence should be addressed to Wenguang Yu yuwgsdufeeducn

Received 1 February 2019 Revised 1 June 2019 Accepted 24 June 2019 Published 7 July 2019

Academic Editor Filippo Cacace

Copyright copy 2019 YunyunWang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we consider the compound Poisson risk model with stochastic premium income We propose a new estimation ofGerber-Shiu function by an efficient method Fourier-cosine series expansion We show that the estimator is easily computed andhas a fast convergence rate Some simulation examples are illustrated to show that the estimation has a good performance when thesample size is finite

1 Introduction

In this paper we consider the following compound Poissonrisk model with stochastic premium income

119880 (119905) = 119906 + 119872(119905)sum119894=1

119884119894 minus 119873(119905)sum119895=1

119883119895 119905 ge 0 (1)

where 119906 ge 0 is the initial surplus and 119880(119905) denotes thesurplus level of an insurance company at time 119905The premiumnumber process 119872(119905) and the claim number process 119873(119905)are homogenous Poisson processes with intensity 120583 gt 0 and120582 gt 0 respectively The individual claim sizes 1198831 1198832 are positive independent and identically distributed (iid)continuous random variables given by generic randomvariable 119883 with density function 119891 The premium sizes1198841 1198842 are positive iid random variables given by genericrandom variable 119884 with exponential distribution function119892(119910) = 120573119890minus120573119910 119910 120573 gt 0 Throughout this paper we assumethat 119872(119905)119905ge0 119873(119905)119905ge0 119883119894119894ge1 and 119884119895119895ge1 are mutuallyindependent

Whenever the surplus process becomes negative we saythat ruin occurs Defining the ruin time by

120591 = inf 119905 ge 0 119880 (119905) lt 0 (2)

where 120591 = infin if for all 119905 ge 0 119880(119905) gt 0 To avoid that ruinis a certain event suppose that the following condition holdsthroughout this paper

Condition 1 (net profit condition)

120583E [119884] gt 120582E [119883] (3)

The above condition guarantees that the expectation of thesurplus process will always be positive at any time 119905 gt 0 thatis to say the limit of surplus process is almost sure positiveOur research is more meaningful under this assumption

Let 120575 gt 0 be the interest force and define the expecteddiscounted penalty function by

Φ (119906)= E [119890minus120575120591w (119880 (120591minus) |119880 (120591)|) 1(120591ltinfin) | 119880 (0) = 119906]

119906 ge 0(4)

where w [0infin)times [0infin) 997888rarr [0infin) is a measurable penaltyfunction of the surplus before ruin and the deficit at ruin and1119860 is an indicator function of the event 119860 This function wasfirst proposed by Gerber and Shiu [1] to study the time toruin the surplus before ruin and the deficit at ruin in the

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 5071268 18 pageshttpsdoiorg10115520195071268

2 Discrete Dynamics in Nature and Society

classical risk model It is also called Gerber-Shiu function inthe literature For recent research progress on the Gerber-Shiu function we can refer to work by Zhao and Yin [2] YinandWang [3] Shen et al [4] Yin and Yuen [5] Zhao and Yao[6] Li et al [7 8] Dong et al [9] among others Recentlyit has become a standard risk measure in ruin theory sincemany important risk problems can be studied via it by takingdifferent forms of penalty function w and different 120575 Severaltypical examples are presented as follows

(1) If w(119909 119910) equiv 1 Φ is the ruin probability when 120575 = 0and it is the Laplace transform of ruin time when 120575 gt0

(2) If w(119909 119910) = 119909+119910Φ is the expected discounted claimsize causing ruin

(3) If w(119909 119910) = 1(119909le1199091) or w(119909 119910) = 1(119910le1199101) for 1199091 1199101 gt0 Φ is the defective distribution of the deficit atruin when 120575 = 0 and it is the discounted defectivedistribution of the deficit at ruin when 120575 gt 0

(4) If w(119909 119910) = 119910119896 Φ is the 119896th moment of the deficit atruin when 120575 = 0 and it is the discounted 119896th momentof the deficit at ruin when 120575 gt 0

The classical compound Poisson risk model assumesthat the premiums are received by insurance company ata constant rate over time [10ndash15] However the premiumsincome in the real life especially for small insurance com-panies is volatile So it is natural to extend the classical riskmodel by replacing the constant premium income with acompound Poisson process which was firstly suggested inBoucherie et al [16] Since then this risk model has beenwidely studied bymany scholars Boikov [17] derived integralequations and exponential bounds for nonruin probabilityTemnov [18] derived a representation for ruin probabil-ity Assuming that the income rate is 1 defective renewalequations satisfied by the discounted penalty function wererespectively obtained by Bao [19] and Yang and Zhang [20]under different distributions of interclaim times Supposingpremiums and claims follow general compound Poissonprocesses a defective renewal equation satisfied by theGerber-Shiu function was established in Labbe and Sendova[21] Zhang and Yang [2] extended their model by assumingthat a specific dependence structure exists among the claimsizes interclaim times and premium sizes then the Laplacetransforms and defective renewal equations for the Gerber-Shiu function were obtained when the individual premiumsizes are exponentially distributed Besides the risk modelwith stochastic income has also been studied in Zhao and Yin[22] Yu [23 24] Xie and Zou [25] Mishura and Ragulina[26] Cheng and Wang [27] Yang et al [28] Zeng et al [29]Deng et al [30] Wang and Zhang [31] and so on

In all of the above papers it is assumed that the probabilitycharacteristics of the surplus process such as the densitiesof claim size and premium size the Poisson intensities ofclaim process and premium process are known In realitythese characteristics for an insurance company are usuallyunknown but some data information on surplus levels claimand premium numbers and claim and premium sizes canbe obtained Thus in recent years many scholars began to

study the estimation of ruin probability and Gerber-Shiufunction based on those data information For the classicalcompound Poisson risk model with unknown claim sizedistribution and Poisson intensity Politis [32] and Masiello[33] respectively proposed the semiparametric estimators ofnonruin and ruin probabilities respectively Zhang et al [34]presented a nonparametric estimator for ruin probabilityZhang [35] proposed a nonparametric estimation of thefinite time ruin probability Zhang and Su [36] proposed anew nonparametric estimation of Gerber-Shiu function byLaguerre series expansion For the risk model perturbed bya Brownian motion Shimizu [37 38] estimated the adjust-ment coefficient and the Gerber-Shiu function by Laplacetransform respectively Zhang [39] estimated the Gerber-Shiu function by Fourier-Sinc series expansion Su et al [40]proposed an estimator for Gerber-Shiu function by Laguerreseries expansion For the estimation of ruin probability andGerber-Shiu function in the Levy risk model we refer theinterested readers to Wang and Yin [41] Zhang and Yang[42 43] Wang et al [44] Shimizu and Zhang [45] Peng andWang [46] and Zhang and Su [47]

In addition Fang and Oosterlee [48] proposed a novelmethod for pricing European options by Fourier-cosineseries expansion This method is also called the COS methodin the literature and it can be easily used to approximatean integrable function as long as the corresponding Fouriertransform has closed-form expression Now the COSmethodhas been widely used for pricing options and other financialderivatives See eg Fang and Oosterlee [49] Ruijter andOosterlee [50] and Zhang and Oosterlee [51] to name a fewRecently theCOSmethod has also been used in risk theory tocompute and estimate some risk measures by some actuarialresearchers For example Chau et al [52 53] used the COSmethod to compute the ruin probability and Gerber-Shiufunction in the Levy riskmodels Zhang [54] applied the COSmethod to compute the density of the time to ruin in theclassical risk model Yang et al [55] used a two-dimensionalCOS method to estimate the discounted density function ofthe deficit at ruin in the classical risk model

Inspired by the work Yang et al [55] in this paper we shalluse the COS method to estimate the Gerber-Shiu functionin the compound Poisson risk model with stochastic incomeWe note that this problem has also been considered by [56]later on where the Laguerre series expansion method is usedto estimate the Gerber-Shiu function In particular it can befound that the convergence rate obtained in Theorem 6 inthis paper is also used by [56] to make error analysis of theirestimator We would like to remark that the convergence ratein Theorem 6 is one of the main contributions of our papersince it plays an important role in studying the consistencyproperty of the estimator

The remainder of this paper is organised as follows InSection 2 we first briefly introduce the Fourier-cosine seriesexpansion method and then derive the Fourier transform ofthe Gerber-Shiu function In Section 3 an estimator of theGerber-Shiu function is proposed by the observed sampleof the surplus process The consistent property is studiedin Section 4 under large sample size setting Finally inSection 5 we present some simulation results to show that

Discrete Dynamics in Nature and Society 3

the estimator behaves well under finite sample size set-ting

2 Preliminaries

21 The Fourier Transform of Gerber-Shiu Function In thissubsection we derive the Fourier transform of the Gerber-Shiu function Let1198711(R+) and 1198712(R+) denote the class of inte-grable and square integrable functions on the positive axisrespectively Let Fℎ and Lℎ denote the Fourier transformand Laplace transform of a ℎ isin 1198711(R+) For any complexnumber 119911 we denote its real part and imaginary part by Re(119911)and Im(119911) respectively

For convenience we introduce the Dickson-Hipp opera-tor 119879119904 (see eg Dickson and Hipp [57] and Li and Garrido[58]) which for any integrable function ℎ on (0infin) and anycomplex number 119904 with Re(119904) ge 0 is defined as

119879119904ℎ (119910) = intinfin119910

119890minus119904(119909minus119910)ℎ (119909) 119889119909 = intinfin0

119890minus119904119909ℎ (119909 + 119910) 119889119909119910 ge 0

(5)

The Dickson-Hipp operator has been widely used in ruintheory to simplify the expression of ruin related functionsFor properties on this operator we refer the interested readersto Li and Garrido [58]

ByTheorem 58 in Labbe and Sendova [21] we know thatwhen E[1198832] lt infin the Gerber-Shiu function Φ satisfies thefollowing renewal equation

Φ(119906) = int1199060Φ (119906 minus 119909) 119891120575 (119909) 119889119909 + 119867 (119906) (6)

where

119891120575 (119909) = 120582120582 + 120583 + 120575 [119891 (119909) + (120573 minus 120588) 119879120588119891 (119909)] 119867 (119906) = 120582120582 + 120583 + 120575 [120596 (119906) + (120573 minus 120588) 119879120588120596 (119906)] 120596 (119906) = intinfin

119906w (119906 119909 minus 119906) 119891 (119909) 119889119909

(7)

It follows from Lemma 53 in Labbe and Sendova [21] that 120588appearing in the above formulae is the unique nonnegativeroot of the following equation (wrt 119904) known as theLundbergrsquos fundamental equation

[120582 + 120583 + 120575 minus 120582L119891 (119904)] (120573 minus 119904) minus 120583120573 = 0 (8)

Remark 1 Let

120594 (119904) = [120582 + 120583 + 120575 minus 120582L119891 (119904)] (119904 minus 120573) + 120583120573 119904 ge 0 (9)

It is clear that 120594(119904) is a continuous function such that 120594(0) =minus120575120573 le 0 120594(120573) = 120583120573 gt 0 and if 120575 gt 0 120594(0) lt 0

In addition since 120583E[119884] gt 120582E[119883] and E[119884] = 1120573 wehave

1205941015840 (119904) = 120582 + 120583 + 120575 minus 120582L119891 (119904) minus 120582119904L1198911015840 (119904)+ 120582120573L1198911015840 (119904)

= 120582 + 120583 + 120575 minus 120582intinfin0

119890minus119904119909119891 (119909) 119889119909+ 120582intinfin

0119904119909119890minus119904119909119891 (119909) 119889119909 minus 120582120573intinfin

0119909119890minus119904119909119889119909

ge 120583 + 120575 minus 120582120573E [119883] gt 0

(10)

which shows that 120594(119904) is an increasing function Thus weconclude that120588 is the unique nonnegative root of the equation120594(119904) = 0 and it is located in the interval [0 120573) In particularwe have 120588 = 0 when 120575 = 0

We assume the following conditions hold true in ourliterature which could be satisfied by penalty functions

Condition 2 (the integrability of w) For the penalty functionw it satisfies

intinfin0

intinfin0

(1 + 119909)w (119909 119910) 119891 (119909 + 119910) 119889119910119889119909 lt infin (11)

Condition 3 For the penalty function w there exist someintegers 1205721 1205722 and constant C such that

w (119909 119910) le 119862 (1 + 119909)1205721 (1 + 119910)1205722 (12)

Theorem 2 Condition 2 guarantees the existence of theFourier transform ofΦ and under Conditions 2 and 3 we haveΦ isin 1198711(R+) cap 1198712(R+)Proof It is easily seen that 119891 119879120588(119891) isin 1198712(R+) UnderCondition 2 we have 120596 119879120588120596 isin 1198711(R+) and

sup119906ge0

119879120588120596 (119906) le intinfin0

intinfin0

w (119909 119910) 119891 (119909 + 119910) 119889119910119889119909lt infin

(13)

thus

intinfin0

(119879120588120596 (119906))2 119889119906 le sup119906ge0

119879120588120596 (119906) times intinfin0

119879120588120596 (119906) 119889119906lt infin

(14)

Furthermore under Condition 3 we have

intinfin0

(120596 (119906))2 119889119906 = sup119906ge0

120596 (119906) times intinfin0

120596 (119906) 119889119906le 119862E [1198831205721+1205722] times intinfin

0120596 (119906) 119889119906 lt infin

(15)

so120596 isin 1198712(R+) and then119867119891120575 isin 1198712(R+) ByTheorem 145 inStenger [59] we haveint119906

0Φ(119906minus119909)119891120575(119909)119889119909 isin 1198712(R+) Using the

inequality (119909 + 119910)2 le 21199092 + 21199102 we finally deriveΦ isin 1198712(R+)due to (6)


Now we compute the Fourier transform of the Gerber-Shiu function Applying the Fourier transform on both sidesof (6) gives

FΦ (119904) = intinfin0

119890119894119904119906Φ(119906) 119889119906= intinfin

0119890119894119904119906 int119906

0Φ (119906 minus 119909) 119891120575 (119909) 119889119909119889119906

+ intinfin0

119890119894119904119906119867(119906) 119889119906= intinfin

0intinfin0

119890119894119904(119911+119909)Φ(119911) 119889119911119889119909 +F119867(119904)= FΦ (119904)F119891120575 (119904) +F119867(119904)

(16)

leading to

FΦ (119904) = F119867(119904)1 minusF119891120575 (119904) (17)

For the Fourier transform F119867(119904) we haveF119867(119904) = intinfin

0119890119894119904119906119867(119906) 119889119906

= 120582120582 + 120583 + 120575 [intinfin0

119890119894119904119906120596 (119906) 119889119906+ (120573 minus 120588)intinfin

0119890119894119904119906119879120588120596 (119906) 119889119906]

= 120582120582 + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)intinfin

0119890119894119904119906119879120588120596 (119906) 119889119906]

(18)

where

intinfin0

119890119894119904119906119879120588120596 (119906) 119889119906 = intinfin0

119890119894119904119906 intinfin119906

119890minus120588(119909minus119906)120596 (119909) 119889119909119889119906= intinfin

0intinfin119906

119890(119894119904+120588)119906 sdot 119890minus120588119909120596 (119909) 119889119909119889119906= intinfin

0119890minus120588119910120596 (119910)int119910

0119890(119894119904+120588)119906119889119906119889119910

= 1120588 + 119894119904 (F120596 (119904) minusL120596 (120588))

(19)

Then we obtain

F119867(119904) = 120582120582 + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)120588 + 119894119904 (F120596 (119904) minusL120596 (120588))]

(20)

Similarly forF119891120575(119904) we obtainF119891120575 (119904) = 120582120582 + 120583 + 120575 [F119891 (119904)

+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minusL119891 (120588))] (21)

22 Fourier-Cosine Series Expansion In this subsection wepresent some known results on the Fourier-cosine seriesexpansion method and give the estimating formula of theGerber-Shiu function derived by Fourier-cosine series expan-sion

By [48] any real function has a cosine expansion whenit is finitely supported Therefore for a integrable functionℎ defined on [1198861 1198862] it has the following cosine seriesexpansion

ℎ (119909) = infinsum119896=0

1015840 21198862 minus 1198861 int1198862

1198861

ℎ (119909) cos(119896120587 119909 minus 11988611198862 minus 1198861)119889119909sdot cos(119896120587 119909 minus 11988611198862 minus 1198861)

(22)

where sum1015840 means the first term of the summation has half

weight For a function ℎ isin 1198711(R+) we introduce an auxiliaryfunction

ℎ119886 (119909) = ℎ (119909) sdot 1(0le119909le119886) 119886 gt 0 (23)

then ℎ119886 has a finite domain [0 119886] and ℎ(119909) = ℎ119886(119909) when119909 isin [0 119886] By (22) we haveℎ (119909) = ℎ119886 (119909)

= infinsum119896=0

1015840 2119886 int1198860ℎ (119909) cos(119896120587119909119886)119889119909 cos (119896120587119909119886)

= infinsum119896=0

1015840 2119886 Reint1198860ℎ (119909) 119890119894(119896120587119886)119909119889119909 cos (119896120587119909119886)

0 le 119909 le 119886

(24)

Due to the existence of the Fourier transform of ℎ theintegrand of Fℎ(sdot) has to decay to zero at +infin and we cantruncate the integration range with a large 119886 then we have

Fℎ (119896120587119886 ) = int+infin0

ℎ (119909) 119890119894(119896120587119886)119909119889119909asymp int119886

0ℎ (119909) 119890119894(119896120587119886)119909119889119909

(25)

Thus

ℎ (119909) asymp infinsum119896=0

1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(26)


Furthermore for a large integer 119870 the above summation canbe truncated as follows

ℎ (119909) asymp 119870minus1sum119896=0

1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(27)

We now consider the Gerber-Shiu function Φ It followsfrom (27) that for 0 le 119906 le 119886

Φ (119906) asymp Φ119870119886 (119906)fl

119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) (28)

We can easily see that the key of approximating the Gerber-Shiu function by Fourier-cosine series expansion method liesin the calculation of the Fourier transform FΦ(119904) for 119904 isin119896120587119886 119896 = 0 1 119870 minus 1 Therefore we derive a specificexpression of Fourier transform FΦ(sdot) in the next section

3 Estimation Procedure

In this section we study how to estimate the Gerber-Shiufunction by Fourier-cosine series expansion based on thediscretely observed information of the surplus process theaggregate claims and premiums processes According to (28)we know that the key is to construct an estimation ofFΦ(119904)based on this discrete information In reality for an insurancecompany even though the relevant probability characteristicsare unknown it is easy to obtain the data sets of surplus levelsthe sizes of claims and premiums and the number of claimsand premiums through observation

Assume that we can observe the surplus process over along time interval [0 119879] LetΔ gt 0 be a fixed interobservationinterval Furthermore wlog we assume 119879Δ is an integerdenoted as 119899

Suppose that the insurer can get the following data sets

(1) Data set of surplus levels

119880119895Δ 119895 = 0 1 2 119899 (29)

where119880119895Δ is the observed surplus level at time 119905 = 119895Δ(2) Data set of claim numbers and claim sizes

119873119895Δ 1198831 1198832 119883119873119895Δ 119895 = 1 119899 (30)

where119873119895Δ is the total claimnumber up to time 119905 = 119895Δ(3) Data set of premium numbers and claim sizes

119872119895Δ 1198841 1198842 119884119873119895Δ 119895 = 1 119899 (31)

where 119872119895Δ is the total premium number up to time119905 = 119895ΔNext we study how to estimate the Fourier transform

FΦ(119904) based on the above data sets To estimate FΦ(119904)

by (17) (20) and (21) we should first estimate the followingcharacteristics

120582 120583 120588 120573 F119891 F120596 L119891(120588) L120596 (120588) (32)

First we can estimate F119891(119904) by the empirical characteristicfunction

F119891 (119904) = 1119873119879

119873119879sum119895=1

119890119894119904119883119895 (33)

SimilarlyL119891(119904) can be estimated by

L119891 (119904) = 1119873119879

119873119879sum119895=1

119890minus119904119883119895 (34)

Next for the function 120596(119906) we haveF120596 (119904) = intinfin

0119890119894119904119906 intinfin

119906w (119906 119909 minus 119906) 119889119909119889119906

= intinfin0

int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909

= E(int1198830119890119894119904119906w (119906 119883 minus 119906) 119889119906)

(35)

Similarly

L120596 (119904) = E(int1198830119890minus119904119906w (119906 119883 minus 119906) 119889119906) (36)

ThenF120596(119904) andL120596(119904) can be respectively estimated by

F120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890119894119904119906w (119906 119883119895 minus 119906) 119889119906

L120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890minus119904119906w (119906119883119895 minus 119906) 119889119906(37)

According to the property of Poisson distribution 120582 and 120583can be estimated by

= 1119879119873119879120583 = 1119879119872119879

(38)

It is easily seen that

minus 120582 = 119874119901 (119879minus12) 120583 minus 120583 = 119874119901 (119879minus12) (39)

Since the premium size 119884 follows exponential distributionwith parameter 120573 we have E[119884] = 1120573 then we can estimate120573 by

120573 = 1(1119872119879)sum119872119879119894=1 119884119894 (40)


It is also easily seen that 120573minus120573 = 119874119901(119879minus12)The estimator of 120588denoted as 120588 is defined to be the positive root of the followingequation (in 119904)

[ + 120583 + 120575 minus L119891 (119904)] (120573 minus 119904) minus 120583120573 = 0 (41)

Since 120588 = 0 as 120575 = 0 we set 120588 = 0 as 120575 = 0Furthermore we can estimate L119891(120588) L120596(120588) respectivelyby L119891(120588) L120596(120588)Remark 3 Let

120594 (119904) = [ + 120583 + 120575 minus L119891 (119904)] (119904 minus 120573) + 120583120573 (42)

It is clear that 120594(0) = minus120575120573 le 0 120594(120573) = 120583120573 gt 0 and if 120575 =0 120594(0) lt 0 In addition since 120583E[119884] gt 120582E[119883] and E[119884] =1120573 it follows from the strong law of large numbers that forany 119904 gt 0

1205941015840 (119904) = + 120583 + 120575 minus L119891 (119904) minus 119904L1198911015840 (119904)+ 120573L1198911015840 (119904)

= + 120583 + 120575 minus 1119873119879

119873119879sum119895=1

119890minus119904119883119895

+ 1119873119879

119873119879sum119895=1

119904119883119895119890minus119904119883119895

minus 120573 1119873119879

119873119879sum119895=1

119883119895119890minus119904119883119895

ge 120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895

119886119904997888997888rarr120583 minus 120582120573E [119883] gt 0

(43)

which shows that the probability that 120594(119904) = 0 has a uniquepositive root 120588 tends to one as 119879 997888rarr infin Thus we concludethat 120588 is located in the interval [0 120573)with probability tendingto one as 119879 997888rarr infin

Proposition 4 Suppose that Condition 1 holds true then wehave 120588 119901997888rarr 120588Proof By Remark 1 120594(119904) is an increasing function and 120594(119904) =0 has unique nonnegative root 120588 Then for any 120576 gt 0 we have120594(120588 minus 120576) lt 0 lt 120594(120588 + 120576) In addition by Remark 3 120594(119904) isnondecreasing on event 120583 gt 120573(1119873119879) sum119873119879119895=1119883119895 and P(120583 gt120573(1119873119879) sum119873119879119895=1119883119895) 997888rarr 1 Also we find that for any 119904 gt 0120594(119904) 119901997888rarr 120594(119904) Thus it follows from Lemma 510 in Van derVaart [60] that 120588 119901997888rarr 120588

Once we have obtained the estimation of the abovecharacteristics by (17) (20) and (21) the estimation of

Fourier transform FΦ(119904) denoted as FΦ(119904) can be definedby

FΦ(119904) = F119867(119904)1 minus F119891120575 (119904) (44)

where

F119867(119904) = + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)

120588 + 119894119904 (F120596 (119904) minus L120596 (120588))]

F119891120575 (119904) = + 120583 + 120575 [F119891 (119904)

+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minus L119891(120588))]

(45)

Finally replacing FΦ(sdot) in (28) by its estimation FΦ(sdot) theGerber-Shiu function can be estimated by

Φ119870119886 (119906) fl 119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) 0 le 119906 le 119886

(46)

4 Asymptotic Properties

In this section we study the asymptotic properties of theestimation Φ119870119886 For any function ℎ isin 1198712(R+) its 1198712-norm isdefined by ℎ = (intinfin

0ℎ2(119909)119889119909)12Throughout this section119862

represents a positive generic constant that may take differentvalues at different steps For two nonnegative functions ℎ1and ℎ2 with common domain Θ sub R+ ℎ1 ≲ ℎ2 means thatℎ1(119909) le 119862ℎ2(119909) uniformly for every 119909 isin Θ In addition wedefine

119867119895 (119909) = int1199090119906119895w (119906 119909 minus 119906) 119889119906 119895 = 0 1 2 (47)

It is easy to see that

intinfin0

119906119895120596 (119906) 119889119906 = E [119867119895 (119883)] (48)

For readerrsquos convenience we introduce some definitionsin empirical process theory which are used to study theasymptotic properties For any measurable function ℎ its119871119903(119875)-norm is defined by ℎ119875119903 = (int |ℎ(120596)|119903119889119875(120596))1119903Given two functions 119897 and 119906 the bracket [119897 119906] is the setof all functions ℎ with 119897 le ℎ le 119906 An 120576-bracket in119871119903(119875) is a bracket [119897 119906] with 119906 minus 119897119875119903 lt 120576 For a classG sub 119871119903(119875) the bracketing number 119873(120576G 119871119903(119875)) is theminimum number of 120576-brackets needed to cover G For 120573 gt0 the bracketing integral is defined by 119869(120573G 119871119903(119875)) =int1205730radic119873(120576G 119871119903(119875))119889120576


We shall use the 1198712-norm to study the asymptotic prop-erties of the estimator

Put Φ119870119886 = Φ119870119886 = 0 when 119906 gt 119886 By triangle inequalitywe have

10038171003817100381710038171003817Φ minus Φ119870119886

10038171003817100381710038171003817 le 1003817100381710038171003817Φ minus Φ1198701198861003817100381710038171003817 + 10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

10038171003817100381710038171003817 (49)

where the first term ΦminusΦ119870119886 is the bias caused by Fourier-cosine series approximation and the second term Φ119870119886 minusΦ119870119886 is the bias caused by statistical estimation

For the bias ΦminusΦ119870119886 by the similar arguments in Zhang[54] we obtain the following result

Theorem 5 Suppose that intinfin0

|Φ1015840(119906)|119889119906 lt infin and for someinteger m Φ(119906) le 119862119906minus(119898+1) then under Conditions 1 2 and 3we have

1003817100381710038171003817Φ minus Φ1198701198861003817100381710038171003817 le 119862 119870 + 11198862119898+1 + 119886119870 minus 1 (50)

Next we study the error Φ119870119886 minus Φ119870119886 For the estimator120588 we derive the following resultTheorem 6 Suppose that Condition 1 holds and E[1198832] lt infinthen we have

120588 minus 120588 = 119874119901 (119879minus12) (51)

Proof By the mean value theorem we have

120594 (120588) = 120594 (120588) + 1205941015840 (120588lowast) (120588 minus 120588) = 1205941015840 (120588lowast) (120588 minus 120588) (52)

where 120588lowast is a random number between 120588 and 120588 Since 120594(120588) =0 we obtain120588 minus 120588 = 120594 (120588) minus 120594 (120588)1205941015840 (120588lowast) (53)

where

120594 (120588) minus 120594 (120588) = [120582 + 120583 + 120575 minus 120582L119891 (120588)] (120588 minus 120573)minus [ + 120583 + 120575 minus L119891 (120588)] (120588 minus 120573)

= (120582 minus ) (120588 minus 120573) (1 minus L119891 (120588))+ (120573 minus 120573) [120575 minus (L119891 (120588) minus 1)]+ 120582 (120588 minus 120573) (L119891 (120588) minusL119891 (120588))+ 120588 (120583 minus 120583)

(54)

Since minus 120582 = 119874119901(119879minus12) 120583 minus 120583 = 119874119901(119879minus12) and 120573 minus 120573 =119874119901(119879minus12) it is easily seen that 120594(120588) minus 120594(120588) = 119874119901(119879minus12)In addition we introduce the following set

119860119879 = 100381610038161003816100381610038161205941015840 (120588lowast)10038161003816100381610038161003816 gt 12 (120583 minus 120582120573E [119883]) (55)

Since 120583 minus 120573(1119873119879) sum119873119879119895=1119883119895

119901997888rarr 120583 minus 120582120573 and |1205941015840(120588lowast)| ge 120583 minus120573(1119873119879)sum119873119879119895=1119883119895 we have

P (119860119879) ge P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 ge 12 (120583 minus 120582120573E [119883]))

= 1 minus P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 lt 12 (120583 minus 120582120573E [119883]))

= 1 minus P(120583 minus 120582120573E [119883] minus 120583 + 120573 1119873119879

119873119879sum119895=1

119883119895

ge 12 (120583 minus 120582120573E [119883])) 997888rarr 1 119886119904 119879 997888rarr infin

(56)

Furthermore

P (1003816100381610038161003816120588 minus 1205881003816100381610038161003816 gt 119862119879minus12)= P(1003816100381610038161003816120594 (120588) minus 120594 (120588)100381610038161003816100381610038161003816100381610038161205941015840 (120588lowast)1003816100381610038161003816 gt 119862119879minus12)le P(1003816100381610038161003816120594 (120588) minus 120594 (120588)100381610038161003816100381610038161003816100381610038161205941015840 (120588lowast)1003816100381610038161003816 gt 119862119879minus12 cap 119860119879)

+ P (119860119888119879)= P (1003816100381610038161003816120594 (120588) minus 120594 (120588)1003816100381610038161003816 gt 119862119879minus12) + P (119860119888119879)

(57)

As a result since 120594(120588) minus 120594(120588) = 119874119901(119879minus12) and P(119860119888119879) 997888rarr 0we derive that 120588 minus 120588 = 119874119901(119879minus12)

The following two theorems give the uniform conver-gence rates ofF119867 andF119891120575Theorem 7 Suppose that Condition 1 holds 119867119895(119883)1198751 ltinfin 119895 = 0 1 and 119867119895(119883)1198752 lt infin 119895 = 1 2 Then for large119886 119870 and 119879 we have

sup119904isin[0119870120587119886]

10038161003816100381610038161003816F119867(119904) minus F119867(119904)10038161003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (58)

Proof By (20) and (44)

F119867(119904) minusF119867(119904)= + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]

minus 120582120582 + 120583 + 120575 (120573 minus 120588)120588 + 119894119904 [F120596 (119904) minusL120596 (120588)]fl 1198681 + 1198682

(59)


where

1198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904) 1198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]

minus 120582120582 + 120583 + 120575 (120573 minus 120588)120588 + 119894119904 [F120596 (119904) minusL120596 (120588)] (60)

We first study 11986811198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904)

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[int1198831198950

119890119894119904119906w (119906 119883119895 minus 119906) 119889119906minus E(int119883

0119890119894119904119906w (119906119883 minus 119906) 119889119906)] + ( + 120583 + 120575

minus 120582120582 + 120583 + 120575)E(int1198830119890119894119904119906w (119906 119883 minus 119906) 119889119906)

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

(61)

where

1198921119904 (119909) = int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906 119909 ge 0 (62)

For 1198921119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E (1198921119904 (119883))1003816100381610038161003816= sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816E(int1198830119890119894119904119906w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816

le 100381610038161003816100381610038161003816100381610038161003816E(int119883

0w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816 = 10038171003817100381710038171198670 (119883)10038171003817100381710038171198751 lt infin

(63)

then

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816(

+ 120583 + 120575 minus 120582120582 + 120583 + 120575)1003816100381610038161003816100381610038161003816100381610038161003816 sdot10038171003817100381710038171198670 (119883)10038171003817100381710038171199011

= 119874119901 (119879minus12)(64)

due to minus 120582 = 119874119901(119879minus12) and 120583 minus 120583 = 119874119901(119879minus12)

Nowwe introduce the following two classes of real-valuedfunctions

G119870119877 = 119892 119892 = Re (1198921119904) 119904 isin [0 119870120587119886 ] G119870119868 = 119892 119892 = Im (1198921119904) 119904 isin [0 119870120587119886 ] (65)

Then we have

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

+ sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

(66)

We only study the convergence rate of the first termsup119892isinG119870119877 |(1119873119879) sum119873119879119895=1[1198921119904(119883119895) minus E(1198921119904(119883))]| since the sec-ond term follows similarly

For any real-valued function 119892 isin G119870119877 we have

1003816100381610038161003816119892 (119909)1003816100381610038161003816 le sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816int119909

0119890119894119904119906w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090w (119906 119909 minus 119906) 119889119906 = 1198670 (119909)

(67)

which implies that G119870119877 is contained in the single bracket[minus1198670 1198670] For two functions 11989211199041 11989211199042 where 119904119895 isin[0119870120587119886] 119895 = 1 2 the mean value theorem gives10038161003816100381610038161003816Re (11989211199041) minus Re (11989211199042)10038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816int

119909

0(cos (1199041119906) minus cos (1199042119906))w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

= 10038161003816100381610038161003816100381610038161003816minus int119909

0sin (119904lowast119906) sdot 119906 sdot (1199041 minus 1199042)w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090119906w (119906 119909 minus 119906) 119889119906 sdot 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

= 1198671 (119909) 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

(68)

where 119904lowast is a number between 1199041 and 1199042 Under the condition1198671(119883)1199012 lt infin it follows from Example 197 in Van derVaart [60] that for any 0 lt 120576 lt 119870120587119886 there exists a constant119862 such that the bracket number forG119870119877 satisfies

119873 (120576G119870119877 1198712 (119875)) le 119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171198752 (69)

As a result for every 120575 gt 0 the bracketing integral119869 (120575G119870119877 1198712 (119875))

le int1205750

radiclog (119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171199012)119889120576 ≲ radiclog (119870119886 )(70)


Furthermore byCorollary 1935 inVanderVaart [60]we have

E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))

≲ radiclog (119870119886)E [radic119873119879] le radiclog (119870119886)

radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)

Combining (66) (73) and (74) we obtain

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)

As a result (64) and (75) give

sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)

Next we consider 1198682 We have

1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]

minus 120582120582 + 120583 + 120575 (120573 minus 120588)120588 + 119894119904 [F120596 (119904) minusL120596 (120588)]= + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950

(119890119894119904119906 minus 119890minus120588119906)w (119906119883119895 minus 119906) 119889119906minus 120582120582 + 120583 + 120575 (120573 minus 120588)120588 + 119894119904sdot E [int119883

0(119890119894119904119906 minus 119890minus120588119906)w (119906 119883 minus 119906) 119889119906]

= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950

(119890minus120588119906 minus 119890minus120588119906)w (119906119883119895 minus 119906) 119889119906

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


0(119890119894119904119906 minus 119890minus120588119906)w (119906 119883 minus 119906) 119889119906] fl 1198971 + 1198972

(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950

(119890119894119904119906 minus 119890minus120588119906)w (119906 119883119895 minus 119906) 119889119906minus 120582120582 + 120583 + 120575 (120573 minus 120588)120588 + 119894119904sdot E [int119883


(78)


For 1198971 by the mean value theorem we have

10038161003816100381610038161003816119890minus120588119906 minus 119890minus12058811990610038161003816100381610038161003816 le 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 119906119890minus120588lowast119906 le 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 119906 (79)

where 120588lowast is a number between 120588 and 120588 Then

100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950

119906 1003816100381610038161003816120588 minus 1205881003816100381610038161003816w (119906119883119895 minus 119906) 119889119906 = + 120583 + 120575sdot (120573 minus 120588)120588 + 119894119904 1003816100381610038161003816120588 minus 1205881003816100381610038161003816 1119873119879

119873119879sum119895=1

1198671 (119883119895)

(80)

Since (1119873119879)sum119873119879119895=11198671(119883119895) 119901997888rarr 1198671(119883)1198751 lt infin 120588 minus 120588 =119874119901(119879minus12) we have

100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


0(119890119894119904119906 minus 119890minus120588119906)w (119906119883 minus 119906) 119889119906]

= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)

sdot ( (120573 minus 120588) + 120583 + 120575 1120588 + 119894119904 minus 120582 (120573 minus 120588)120582 + 120583 + 120575 1120588 + 119894119904)

sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090

119890119894119904119906 minus 119890minus120588119906120588 + 119894119904 w (119906 119909 minus 119906) 119889119906= int119909

0int1199060119890119894119904(119906minus119910)minus120588119910119889119910w (119906 119909 minus 119906) 119889119906 119909 ge 0

(83)

Under condition 1198672(119883)1198752 lt infin by the similar argumentson 1198921119904 we conclude that

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have

(120588 + 119894119904)( (120573 minus 120588) + 120583 + 120575 1120588 + 119894119904 minus 120582 (120573 minus 120588)120582 + 120583 + 120575 1120588 + 119894119904)

= 119874119901 (119879minus12)(85)

due to minus 120582 = 119874119901(119879minus12) 120573 minus 120573 = 119874119901(119879minus12) 120583 minus 120583 =119874119901(119879minus12) and 120588 minus 120588 = 119874119901(119879minus12) For 1198922119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin

0int1199090int1199060119890minus120588119910119889119910w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909

le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)

By (84) and (87) we derive that

sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)

Furthermore (81) and (88) give

sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)


Combining (76) and (89) we eventually obtain the desiredresult

Theorem8 Suppose that Condition 1 holds E[119883119896] lt infin 119896 =1 2 3 4 Then for large 119886 119870 and 119879 we havesup

119904isin[0119870120587119886]

10038161003816100381610038161003816F119891120575 (119904) minus F119891120575 (119904)10038161003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (90)


F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


First we study 1198681198681 Note that1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)

By the similar arguments on 1198921119904 when 119864[119883] lt infin weconclude that

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)

Then (96) and (97) give

sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)

Next we study 1198681198682 Note that1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus 119890minus120588119883119895]minus 120582120582 + 120583 + 120575 (120573 minus 120588)120588 + 119894119904 E [119890119894119904119883 minus 119890minus120588119883]

= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1

[119890minus120588119883119895 minus 119890minus120588119883119895]

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus 119890minus120588119883119895]minus 120582120582 + 120583 + 120575 (120573 minus 120588)120588 + 119894119904 E [119890119894119904119883 minus 119890minus120588119883] fl 1198971198971 + 1198971198972

(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)

By the similar arguments on 1198971 in the proof of Theorem 6when E[119883] lt infin we can derive that

100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus 119890minus120588119883119895]minus 120582120582 + 120583 + 120575 (120573 minus 120588)120588 + 119894119904 E [119890119894119904119883 minus 119890minus120588119883] = + 120583 + 120575sdot (120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]

+ (120588 + 119894119904) ( (120573 minus 120588) + 120583 + 120575 1120588 + 119894119904 minus 120582 (120573 minus 120588)120582 + 120583 + 120575 1120588 + 119894119904)

sdot E (1198924119904 (119883))

(102)

where

1198924119904 (119909) = 119890119894119904119909 minus 119890minus120588119909120588 + 119894119904 = intinfin0

119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)

Under condition E[119883119896] lt infin 119896 = 1 2 3 4 by the similararguments on 1198921119904 we conclude that

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)

In addition it follows from (104) and the similar analysis of1198972 in the proof of Theorem 6 that

sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)

Finally we can derive the desired result by (98) and (106)

Based on the above conclusions we have the followingresult

Theorem9 Suppose that Condition 1 holds E[119883119896] lt infin 119896 =1 2 3 4 119867119895(119883)1198751 lt infin 119895 = 0 1 and 119867119895(119883)1198752 ltinfin 119895 = 1 2 Then for large 119886 119870 and 119879 we have10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860

10038161003816100381610038161003816Φ119870119886 (119906) minus Φ119870119886 (119906)100381610038161003816100381610038162 119889119906le 119870minus1sum119896=0

41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886

sdot 119870minus1sum119896=0

10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)

Then by (17) and (44) we obtain

sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ(119904) minus FΦ(119904)10038161003816100381610038161003816 = sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(1 minusF119891120575 (119904)) (F119867(119904) minus F119867(119904)) +F119867(119904) (F119891120575 (119904) minus F119891120575 (119904))(1 minusF119891120575 (119904)) (1 minus F119891120575 (119904))

10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u

Figure 1 Estimation of the Gerber-Shiu function for exponential claim sizes true value curves (red curves) and 30 estimated curves (greencurves) when (119879 = 52) (a1) Ruin probability (a2) Laplace transform of ruin time (a3) Expected discounted deficit at ruin due to a claim

Combining Theorems 6 and 7 gives

sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ(119904) minus FΦ(119904)10038161003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (110)

Finally by (108) we derive that

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)

This completes the proof

Combining Theorems 5 and 9 we finally obtain thefollowing convergence rate

10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)

We first find the optimal truncation parameter 119886lowast =119874(119870minus(119899+1)) to minimize the convergence rate 119874((119870 +1)1198862119898+1) + 119874(119886(119870 minus 1)) Replacing 119886 by 119886lowast in (112) gives

10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)

In addition we find the optimal truncation 119870lowast =119874(119879(119898+1)2119898) Thus we obtain the smallest convergence rateΦ minus Φ1198701198862 = 119874119901(119879minus21198982(119898+1)2)5 Simulations

In this section we present some simulation examples toillustrate the performance of our estimator when sample isfinite Both the exponential claims with density function119891(119909) = 119890minus119909 119909 gt 0 and the Erlang(2) claims with density

function 119891(119909) = 4119909119890minus2119909 119909 gt 0 are considered in oursimulation We estimate the following three specific Gerber-Shiu functions (1) ruin probability (RP w equiv 1 120575 = 0) (2)Laplace transform of ruin time (LT w equiv 1 120575 = 01) (3)expected discounted deficit at ruin (EDD) when ruin is dueto a claim (w(119909 119910) = 119910 120575 = 01)

Suppose that in a long time interval the insurer willobserve the data once a week thus we setΔ = 1 which can beexplained as one week In all cases we set 120582 = 2 120583 = 5 that isto say the expected claim number is 2 times perweek and theexpected premiumnumber is 5 times perweek Furthermoresince there are 52 business weeks every year we assume that119879 = 1 times 52 times Δ 119879 = 3 times 52 times Δ 119879 = 5 times 52 times Δ 119879 =7 times 52timesΔThenwe use formula (46) to estimate RP LT EDDdue to a claim In all simulations we set 119886 = 30 119870 = 213and we carry out the relevant analysis based on 300 timessimulation results In this section the mean value and themean relative error are respectively defined by

1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)

and the integrated mean square error (IMSE) is defined by

1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)

where Φ119870119886119895(119906) is the estimate of Gerber-Shiu function inthe 119895-th experiment For IMSE we compute the integral onthe finite domain [0 30] in that both the true value and theestimator will be very small when 119906 ge 30

First to show variability bands and illustrate the stabilityof our estimation procedures when 119879 = 52 we plot 30consecutive estimate curves (green curves) of Gerber-Shiufunctions on the same picture together with the associatedtruth curves (red curves) in Figures 1 and 2 In all examples


(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u

Figure 2 Estimation of the Gerber-Shiu function for Erlang(2) claim sizes true value curves (red curves) and 30 estimated curves (greencurves) when (119879 = 52) (b1) Ruin probability (b2) Laplace transform of ruin time (b3) Expected discounted deficit at ruin due to a claim

T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u

Figure 3 Estimation of the Gerber-Shiu function for exponential claim sizes mean curves (c1) Ruin probability (c2) Laplace transform ofruin time (c3) Expected discounted deficit at ruin due to a claim

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u

Figure 4 Estimation of the Gerber-Shiu function for Erlang(2) claim sizes mean curves (d1) Ruin probability (d2) Laplace transform ofruin time (d3) Expected discounted deficit at ruin due to a claim

we observe that all estimate curves are very close and theyhave the same trend with their reference curves This showsthat our estimations can match the real functions perfectlyand they have good stationarity Next in Figures 3 and 4 weshow the mean value curves for different observe intervals(namely different 119879) and compared them with the meanvalue curves It is obvious that our estimations are accurate inthe sense of mean and the mean value curves converge to thetrue value curves as 119879 increases In addition Figures 5 and

6 display the mean relative error curves of our estimationsfor different observe intervals We find that the mean relativeerrors increase with the increase of 119906 They are small when 119906is small however they are very large when 119906 becomes largeWe can also observe that the mean relative errors decrease as119879 increases This trend shows that our estimations performbetter with larger 119879

Finally we compare the Fourier-cosine series expansionmethod with the FFT method used in [45] The parameter


T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u

Figure 5 Estimation of the Gerber-Shiu function for exponential claim sizes mean relative error curves (e1) Ruin probability (e2) Laplacetransform of ruin time (e3) Expected discounted deficit at ruin due to a claim

0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u

Figure 6 Estimation of the Gerber-Shiu function for Erlang(2) claim sizes mean relative error curves (f1) Ruin probability (f2) Laplacetransform of ruin time (f3) Expected discounted deficit at ruin due to a claim

Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u

Figure 7 Comparing Fourier-cosine method with FFT method for exponential claim sizes when 119879 = 52 mean relative error curves (g1)Ruin probability (g2) Laplace transform of ruin time (g3) Expected discounted deficit at ruin due to a claim

setting of FFT is the same as in [45] First we present theIMSE values for both methods in Table 1 and we find that theFourier-cosine series expansion method can lead to smallerIMSEs compared with FFTmethod Besides we use themeanrelative error to compare these two methods when 119879 = 52The results are illustrated in Figures 7 and 8 It is easilyseen that Fourier-cosine series expansion method can yieldsmaller mean relative errors

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest


Table 1 IMSE for the estimation of Gerber-Shiu function

Exponential distribution Erlang(2) distribution119879 RP LT EDD RP LT EDD

Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u

Figure 8 Comparing Fourier-cosine method with FFTmethod for Erlang(2) claim sizes when 119879 = 52 mean relative error curves (h1) Ruinprobability (h2) Laplace transform of ruin time (h3) Expected discounted deficit at ruin due to a claim

Acknowledgments

This research was financially supported by the National Natu-ral Science Foundation of China (no 11301303 no 71804090and no 11501325) the National Social Science Foundationof China (no 15BJY007) the Taishan Scholars Program ofShandong Province (no tsqn20161041) the Humanities andSocial Sciences Project of the Ministry Education of China(no 16YJC630070 and no 19YJA910002) the Natural ScienceFoundation of Shandong Province (no ZR2018MG002) theFostering Project of Dominant Discipline and Talent Teamof Shandong Province Higher Education Institutions (no1716009) the Risk Management and Insurance ResearchTeam of Shandong University of Finance and Economicsthe 1251 Talent Cultivation Project of Shandong JiaotongUniversity the Collaborative Innovation Center Project ofthe Transformation of New and Old Kinetic Energy andGovernment Financial Allocation Shandong Jiaotong Uni-versity ldquoClimbingrdquo Research Innovation Team Program andExcellent Talents Project of Shandong University of Financeand Economics

References

[1] H U Gerber and E S W Shiu ldquoOn the time value of ruinrdquoNorth American Actuarial Journal vol 2 no 1 pp 48ndash78 1998

[2] X H Zhao and C C Yin ldquoThe Gerber-Shiu expected dis-counted penalty function for Levy insurance risk processesrdquo

ActaMathematicae Applicatae Sinica vol 26 no 4 pp 575ndash5862010

[3] C C Yin and C W Wang ldquoThe perturbed compound Poissonrisk process with investment and debit interestrdquo Methodologyand Computing in Applied Probability vol 12 no 3 pp 391ndash4132010

[4] Y Shen C C Yin and K C Yuen ldquoAlternative approach to theoptimality of the threshold strategy for spectrally negative Levyprocessesrdquo Acta Mathematicae Applicatae Sinica vol 23 no 4pp 705ndash716 2013

[5] C C Yin and K C Yuen ldquoExact joint laws associated withspectrally negative Levy processes and applications to insurancerisk theoryrdquo Frontiers of Mathematics in China vol 9 no 6 pp1453ndash1471 2014

[6] Y X Zhao andD J Yao ldquoOptimal dividend and capital injectionproblem with a random time horizon and a ruin penalty inthe dual modelrdquo Applied Mathematics-A Journal of ChineseUniversities vol 30 no 3 pp 325ndash339 2015

[7] S Li Y Lu andK P Sendova ldquoThe expected discounted penaltyfunction from infinite time to finite timerdquo Scandinavian Actu-arial Journal vol 2019 no 4 pp 336ndash354 2019

[8] Y Q Li C C Yin and X W Zhou ldquoOn the last exit timesfor spectrally negative Levy processesrdquo Journal of AppliedProbability vol 54 no 2 pp 474ndash489 2017

[9] H Dong C C Yin andH S Dai ldquoSpectrally negative Levy riskmodel under Erlangized barrier strategyrdquo Journal of Computa-tional and Applied Mathematics vol 351 pp 101ndash116 2019


[10] H Dong and C C Yin ldquoComplete monotonicity of the prob-ability of ruin and de Finettirsquos dividend problemrdquo Journal ofSystems Science amp Complexity vol 25 no 1 pp 178ndash185 2012

[11] CWWang C C Yin and E Q Li ldquoOn the classical risk modelwith credit and debit interests under absolute ruinrdquo Statistics ampProbability Letters vol 80 no 5-6 pp 427ndash436 2010

[12] C C Yin and K C Yuen ldquoOptimality of the threshold dividendstrategy for the compound Poisson modelrdquo Statistics amp Proba-bility Letters vol 81 no 12 pp 1841ndash1846 2011

[13] W G Yu Y J Huang and C R Cui ldquoThe absolute ruin insur-ance risk model with a threshold dividend strategyrdquo Symmetry-Basel vol 10 no 9 pp 377ndash396 2018

[14] WG Yu ldquoSome results on absolute ruin in the perturbed insur-ance risk model with investment and debit interestsrdquo EconomicModelling vol 31 no 1 pp 625ndash634 2013

[15] Y X Zhao R M Wang and D J Yao ldquoOptimal dividend andequity issuance in the perturbed dual model under a penalty forruinrdquo Communications in StatisticsmdashTheory and Methods vol45 no 2 pp 365ndash384 2016

[16] R J BoucherieO J Boxma andK Sigman ldquoA note on negativecustomers GIG1 workload and risk processrdquo Probability inthe Engineering and Informational Sciences vol 11 no 3 pp305ndash311 1997

[17] A V Boikov ldquoThe Cramer-Lundberg model with stochasticpremium processrdquoTheory of Probability amp Its Applications vol47 no 3 pp 489ndash493 2003

[18] G Temnov ldquoRisk process with random incomerdquo Journal ofMathematical Sciences vol 123 no 1 pp 3780ndash3794 2004

[19] Z H Bao ldquoThe expected discounted penalty at ruin in the riskprocess with random incomerdquo Applied Mathematics and Com-putation vol 179 no 2 pp 559ndash566 2006

[20] H Yang and Z Zhang ldquoOn a class of renewal risk model withrandom incomerdquo Applied Stochastic Models in Business andIndustry vol 25 no 6 pp 678ndash695 2009

[21] C Labbe and K P Sendova ldquoThe expected discounted penaltyfunction under a risk model with stochastic incomerdquo AppliedMathematics and Computation vol 215 no 5 pp 1852ndash18672009

[22] Y X Zhao and C C Yin ldquoThe expected discounted penaltyfunction under a renewal risk model with stochastic incomerdquoApplied Mathematics and Computation vol 218 no 10 pp6144ndash6154 2012

[23] W G Yu ldquoRandomized dividends in a discrete insurancerisk model with stochastic premium incomerdquo MathematicalProblems in Engineering vol 2013 Article ID 579534 9 pages2013

[24] W G Yu ldquoOn the expected discounted penalty function for aMarkov regime-switching insurance risk model with stochasticpremium incomerdquoDiscrete Dynamics inNature and Society vol2013 Article ID 320146 9 pages 2013

[25] J H Xie and W Zou ldquoOn a risk model with random incomesand dependence between claim sizes and claim intervalsrdquoIndagationes Mathematicae vol 24 no 3 pp 557ndash580 2013

[26] Y Mishura and O Ragulina Ruin probabilities SmoothnessBounds Supermartingale Approach ISTE Press-Elsevier Lon-don UK 2016

[27] J H Cheng and D H Wang ldquoRuin probabilities for a two-dimensional perturbed risk model with stochastic premiumsrdquoActa Mathematicae Applicatae Sinica vol 32 no 4 pp 1053ndash1066 2016

[28] X X Yang J Y Tan H J Zhang and Z Q Li ldquoAn optimalcontrol problem in a risk model with stochastic premiums andperiodic dividend paymentsrdquo Asia-Pacific Journal of Opera-tional Research vol 34 no 3 Article ID 1740013-1-18 2017

[29] Y Zeng D P Li Z Chen and Z Yang ldquoAmbiguity aversion andoptimal derivative-based pension investment with stochasticincome and volatilityrdquo Journal of Economic Dynamics andControl (JEDC) vol 88 pp 70ndash103 2018

[30] YCDeng J Liu YHuangM Li and JMZhou ldquoOnadiscreteinteraction risk model with delayed claims and stochasticincomes under random discount ratesrdquo Communications inStatisticsmdashTheory and Methods vol 47 no 23 pp 5867ndash58832017

[31] W Y Wang and Z M Zhang ldquoComputing the Gerber-Shiufunction by frame duality projectionrdquo Scandinavian ActuarialJournal vol 2019 no 4 pp 291ndash307 2019

[32] K Politis ldquoSemiparametric estimation for non-ruin probabili-tiesrdquo Scandinavian Actuarial Journal vol 2003 no 1 pp 75ndash962003

[33] E Masiello ldquoOn semiparametric estimation of ruin probabili-ties in the classical risk modelrdquo Scandinavian Actuarial Journalvol 2014 no 4 pp 283ndash308 2014

[34] Z M Zhang H L Yang and H Yang ldquoOn a nonparametricestimator for ruin probability in the classical risk modelrdquoScandinavian Actuarial Journal vol 2014 no 4 pp 309ndash3382014

[35] Z M Zhang ldquoNonparametric estimation of the finite time ruinprobability in the classical risk modelrdquo Scandinavian ActuarialJournal vol 2017 no 5 pp 452ndash469 2017

[36] Z M Zhang andW Su ldquoA new efficient method for estimatingthe Gerber-Shiu function in the classical risk modelrdquo Scandina-vian Actuarial Journal vol 2018 no 5 pp 426ndash449 2018

[37] Y Shimizu ldquoA new aspect of a risk process and its statisticalinferencerdquo Insurance Mathematics and Economics vol 44 no1 pp 70ndash77 2009

[38] Y Shimizu ldquoEstimation of the expected discounted penaltyfunction for Levy insurance risksrdquo Mathematical Methods ofStatistics vol 20 no 2 pp 125ndash149 2011

[39] Z M Zhang ldquoEstimating the gerber-shiu function by Fourier-Sinc series expansionrdquo Scandinavian Actuarial Journal vol2017 no 10 pp 898ndash919 2017

[40] W Su Y D Yong and Z M Zhang ldquoEstimating the Gerber-Shiu function in the perturbed compound Poisson model byLaguerre series expansionrdquo Journal of Mathematical Analysisand Applications vol 469 no 2 pp 705ndash729 2019

[41] Y FWang andCC Yin ldquoApproximation for the ruin probabili-ties in a discrete time riskmodel with dependent risksrdquo Statisticsamp Probability Letters vol 80 no 17-18 pp 1335ndash1342 2010

[42] Z M Zhang and H L Yang ldquoNonparametric estimate of theruin probability in a pure-jump Levy risk modelrdquo InsuranceMathematics and Economics vol 53 no 1 pp 24ndash35 2013

[43] Z M Zhang and H L Yang ldquoNonparametric estimation forthe ruin probability in a Levy risk model under low-frequencyobservationrdquo Insurance Mathematics and Economics vol 59pp 168ndash177 2014

[44] Y-F Wang C-C Yin and X-S Zhang ldquoUniform estimatefor the tail probabilities of randomly weighted sumsrdquo ActaMathematicae Applicatae Sinica vol 30 no 4 pp 1063ndash10722014

[45] Y Shimizu and Z M Zhang ldquoEstimating Gerber-Shiu func-tions from discretely observed Levy driven surplusrdquo InsuranceMathematics and Economics vol 74 pp 84ndash98 2017


[46] J Y Peng and D C Wang ldquoUniform asymptotics for ruinprobabilities in a dependent renewal risk model with stochasticreturn on investmentsrdquo Stochastics An International Journal ofProbability and Stochastic Processes vol 90 no 3 pp 432ndash4712018

[47] Z M Zhang and W Su ldquoEstimating the Gerber-Shiu functionin a Levy risk model by Laguerre series expansionrdquo Journal ofComputational and Applied Mathematics vol 346 pp 133ndash1492019

[48] F Fang and C W Oosterlee ldquoA novel pricing method forEuropean options based on Fourier-cosine series expansionsrdquoSIAM Journal on Scientific Computing vol 31 no 2 pp 826ndash848 2008

[49] F Fang and C W Oosterlee ldquoPricing early-exercise and dis-crete barrier options by fourier-cosine series expansionsrdquoNumerische Mathematik vol 114 no 1 pp 27ndash62 2009

[50] M J Ruijter and C W Oosterlee ldquoTwo-dimensional fouriercosine series expansion method for pricing financial optionsrdquoSIAM Journal on Scientific Computing vol 34 no 5 pp B642ndashB671 2012

[51] B Zhang and C W Oosterlee ldquoPricing of early-exerciseAsian options under Levy processes based on Fourier cosineexpansionsrdquo Applied Numerical Mathematics vol 78 pp 14ndash302014

[52] K W Chau S C P Yam and H Yang ldquoFurier-cosine methodfor ruin probabilitiesrdquo Journal of Computational amp AppliedMathematics vol 281 pp 94ndash106 2015

[53] KW Chau S C Yam andH Yang ldquoFourier-cosinemethod forGerber-Shiu functionsrdquo InsuranceMathematics and Economicsvol 61 pp 170ndash180 2015

[54] Z M Zhang ldquoApproximating the density of the time to ruin viaFourier-cosine series expansionrdquo ASTIN Bulletin vol 41 no 1pp 169ndash198 2017

[55] Y Yang W Su and Z M Zhang ldquoEstimating the discounteddensity of the deficit at ruin by Fourier cosine series expansionrdquoStatistics amp Probability Letters vol 146 pp 147ndash155 2019

[56] W Su B X Shi and Y Y Wang ldquoEstimating the gerber-shiu function under a risk model with stochastic income bylaguerre series expansionrdquo Communications in Statistics-TheoryandMethods httpsdoiorg1010800361092620191620782 inpress

[57] D C M Dickson and C Hipp ldquoOn the time to ruin forErlang(2) risk processesrdquo Insurance Mathematics amp Economicsvol 29 no 3 pp 333ndash344 2001

[58] S Li and J Garrido ldquoOn ruin for the Erlang(n) risk processrdquoInsurance Mathematics and Economics vol 34 no 3 pp 391ndash408 2004

[59] F Stenger Numerical Methods Based on Sinc and AnalyticFunctions Springer-Verlag New York NY USA 1993

[60] A W V D Vaart Asymptotic Statistics Cambridge UniversityPress Cambridge UK 1998

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classical risk model It is also called Gerber-Shiu function inthe literature For recent research progress on the Gerber-Shiu function we can refer to work by Zhao and Yin [2] YinandWang [3] Shen et al [4] Yin and Yuen [5] Zhao and Yao[6] Li et al [7 8] Dong et al [9] among others Recentlyit has become a standard risk measure in ruin theory sincemany important risk problems can be studied via it by takingdifferent forms of penalty function w and different 120575 Severaltypical examples are presented as follows

(1) If w(119909 119910) equiv 1 Φ is the ruin probability when 120575 = 0and it is the Laplace transform of ruin time when 120575 gt0

(2) If w(119909 119910) = 119909+119910Φ is the expected discounted claimsize causing ruin

(3) If w(119909 119910) = 1(119909le1199091) or w(119909 119910) = 1(119910le1199101) for 1199091 1199101 gt0 Φ is the defective distribution of the deficit atruin when 120575 = 0 and it is the discounted defectivedistribution of the deficit at ruin when 120575 gt 0

(4) If w(119909 119910) = 119910119896 Φ is the 119896th moment of the deficit atruin when 120575 = 0 and it is the discounted 119896th momentof the deficit at ruin when 120575 gt 0

The classical compound Poisson risk model assumesthat the premiums are received by insurance company ata constant rate over time [10ndash15] However the premiumsincome in the real life especially for small insurance com-panies is volatile So it is natural to extend the classical riskmodel by replacing the constant premium income with acompound Poisson process which was firstly suggested inBoucherie et al [16] Since then this risk model has beenwidely studied bymany scholars Boikov [17] derived integralequations and exponential bounds for nonruin probabilityTemnov [18] derived a representation for ruin probabil-ity Assuming that the income rate is 1 defective renewalequations satisfied by the discounted penalty function wererespectively obtained by Bao [19] and Yang and Zhang [20]under different distributions of interclaim times Supposingpremiums and claims follow general compound Poissonprocesses a defective renewal equation satisfied by theGerber-Shiu function was established in Labbe and Sendova[21] Zhang and Yang [2] extended their model by assumingthat a specific dependence structure exists among the claimsizes interclaim times and premium sizes then the Laplacetransforms and defective renewal equations for the Gerber-Shiu function were obtained when the individual premiumsizes are exponentially distributed Besides the risk modelwith stochastic income has also been studied in Zhao and Yin[22] Yu [23 24] Xie and Zou [25] Mishura and Ragulina[26] Cheng and Wang [27] Yang et al [28] Zeng et al [29]Deng et al [30] Wang and Zhang [31] and so on

In all of the above papers it is assumed that the probabilitycharacteristics of the surplus process such as the densitiesof claim size and premium size the Poisson intensities ofclaim process and premium process are known In realitythese characteristics for an insurance company are usuallyunknown but some data information on surplus levels claimand premium numbers and claim and premium sizes canbe obtained Thus in recent years many scholars began to

study the estimation of ruin probability and Gerber-Shiufunction based on those data information For the classicalcompound Poisson risk model with unknown claim sizedistribution and Poisson intensity Politis [32] and Masiello[33] respectively proposed the semiparametric estimators ofnonruin and ruin probabilities respectively Zhang et al [34]presented a nonparametric estimator for ruin probabilityZhang [35] proposed a nonparametric estimation of thefinite time ruin probability Zhang and Su [36] proposed anew nonparametric estimation of Gerber-Shiu function byLaguerre series expansion For the risk model perturbed bya Brownian motion Shimizu [37 38] estimated the adjust-ment coefficient and the Gerber-Shiu function by Laplacetransform respectively Zhang [39] estimated the Gerber-Shiu function by Fourier-Sinc series expansion Su et al [40]proposed an estimator for Gerber-Shiu function by Laguerreseries expansion For the estimation of ruin probability andGerber-Shiu function in the Levy risk model we refer theinterested readers to Wang and Yin [41] Zhang and Yang[42 43] Wang et al [44] Shimizu and Zhang [45] Peng andWang [46] and Zhang and Su [47]

In addition Fang and Oosterlee [48] proposed a novelmethod for pricing European options by Fourier-cosineseries expansion This method is also called the COS methodin the literature and it can be easily used to approximatean integrable function as long as the corresponding Fouriertransform has closed-form expression Now the COSmethodhas been widely used for pricing options and other financialderivatives See eg Fang and Oosterlee [49] Ruijter andOosterlee [50] and Zhang and Oosterlee [51] to name a fewRecently theCOSmethod has also been used in risk theory tocompute and estimate some risk measures by some actuarialresearchers For example Chau et al [52 53] used the COSmethod to compute the ruin probability and Gerber-Shiufunction in the Levy riskmodels Zhang [54] applied the COSmethod to compute the density of the time to ruin in theclassical risk model Yang et al [55] used a two-dimensionalCOS method to estimate the discounted density function ofthe deficit at ruin in the classical risk model

Inspired by the work Yang et al [55] in this paper we shalluse the COS method to estimate the Gerber-Shiu functionin the compound Poisson risk model with stochastic incomeWe note that this problem has also been considered by [56]later on where the Laguerre series expansion method is usedto estimate the Gerber-Shiu function In particular it can befound that the convergence rate obtained in Theorem 6 inthis paper is also used by [56] to make error analysis of theirestimator We would like to remark that the convergence ratein Theorem 6 is one of the main contributions of our papersince it plays an important role in studying the consistencyproperty of the estimator

The remainder of this paper is organised as follows InSection 2 we first briefly introduce the Fourier-cosine seriesexpansion method and then derive the Fourier transform ofthe Gerber-Shiu function In Section 3 an estimator of theGerber-Shiu function is proposed by the observed sampleof the surplus process The consistent property is studiedin Section 4 under large sample size setting Finally inSection 5 we present some simulation results to show that



2 Preliminaries



119879119904ℎ (119910) = intinfin119910


119890minus119904119909ℎ (119909 + 119910) 119889119909119910 ge 0

(5)



Φ(119906) = int1199060Φ (119906 minus 119909) 119891120575 (119909) 119889119909 + 119867 (119906) (6)

where


119906w (119906 119909 minus 119906) 119891 (119909) 119889119909

(7)



Remark 1 Let

120594 (119904) = [120582 + 120583 + 120575 minus 120582L119891 (119904)] (119904 minus 120573) + 120583120573 119904 ge 0 (9)



1205941015840 (119904) = 120582 + 120583 + 120575 minus 120582L119891 (119904) minus 120582119904L1198911015840 (119904)+ 120582120573L1198911015840 (119904)

= 120582 + 120583 + 120575 minus 120582intinfin0

119890minus119904119909119891 (119909) 119889119909+ 120582intinfin


0119909119890minus119904119909119889119909

ge 120583 + 120575 minus 120582120573E [119883] gt 0

(10)




intinfin0

intinfin0

(1 + 119909)w (119909 119910) 119891 (119909 + 119910) 119889119910119889119909 lt infin (11)


w (119909 119910) le 119862 (1 + 119909)1205721 (1 + 119910)1205722 (12)


sup119906ge0

119879120588120596 (119906) le intinfin0

intinfin0

w (119909 119910) 119891 (119909 + 119910) 119889119910119889119909lt infin

(13)

thus

intinfin0

(119879120588120596 (119906))2 119889119906 le sup119906ge0

119879120588120596 (119906) times intinfin0

119879120588120596 (119906) 119889119906lt infin

(14)


intinfin0

(120596 (119906))2 119889119906 = sup119906ge0



0120596 (119906) 119889119906 lt infin

(15)







119890119894119904119906Φ(119906) 119889119906= intinfin

0119890119894119904119906 int119906

0Φ (119906 minus 119909) 119891120575 (119909) 119889119909119889119906

+ intinfin0

119890119894119904119906119867(119906) 119889119906= intinfin

0intinfin0

119890119894119904(119911+119909)Φ(119911) 119889119911119889119909 +F119867(119904)= FΦ (119904)F119891120575 (119904) +F119867(119904)

(16)

leading to

FΦ (119904) = F119867(119904)1 minusF119891120575 (119904) (17)


0119890119894119904119906119867(119906) 119889119906

= 120582120582 + 120583 + 120575 [intinfin0

119890119894119904119906120596 (119906) 119889119906+ (120573 minus 120588)intinfin

0119890119894119904119906119879120588120596 (119906) 119889119906]

= 120582120582 + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)intinfin

0119890119894119904119906119879120588120596 (119906) 119889119906]

(18)

where

intinfin0

119890119894119904119906119879120588120596 (119906) 119889119906 = intinfin0

119890119894119904119906 intinfin119906


0intinfin119906


0119890minus120588119910120596 (119910)int119910

0119890(119894119904+120588)119906119889119906119889119910

= 1120588 + 119894119904 (F120596 (119904) minusL120596 (120588))

(19)

Then we obtain

F119867(119904) = 120582120582 + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)120588 + 119894119904 (F120596 (119904) minusL120596 (120588))]

(20)


+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minusL119891 (120588))] (21)



ℎ (119909) = infinsum119896=0

1015840 21198862 minus 1198861 int1198862

1198861


(22)



ℎ119886 (119909) = ℎ (119909) sdot 1(0le119909le119886) 119886 gt 0 (23)


= infinsum119896=0

1015840 2119886 int1198860ℎ (119909) cos(119896120587119909119886)119889119909 cos (119896120587119909119886)

= infinsum119896=0

1015840 2119886 Reint1198860ℎ (119909) 119890119894(119896120587119886)119909119889119909 cos (119896120587119909119886)

0 le 119909 le 119886

(24)


Fℎ (119896120587119886 ) = int+infin0

ℎ (119909) 119890119894(119896120587119886)119909119889119909asymp int119886

0ℎ (119909) 119890119894(119896120587119886)119909119889119909

(25)

Thus


1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(26)



ℎ (119909) asymp 119870minus1sum119896=0

1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(27)


Φ (119906) asymp Φ119870119886 (119906)fl

119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) (28)







119880119895Δ 119895 = 0 1 2 119899 (29)


119873119895Δ 1198831 1198832 119883119873119895Δ 119895 = 1 119899 (30)


119872119895Δ 1198841 1198842 119884119873119895Δ 119895 = 1 119899 (31)




120582 120583 120588 120573 F119891 F120596 L119891(120588) L120596 (120588) (32)


F119891 (119904) = 1119873119879

119873119879sum119895=1

119890119894119904119883119895 (33)


L119891 (119904) = 1119873119879

119873119879sum119895=1

119890minus119904119883119895 (34)


0119890119894119904119906 intinfin

119906w (119906 119909 minus 119906) 119889119909119889119906

= intinfin0

int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909

= E(int1198830119890119894119904119906w (119906 119883 minus 119906) 119889119906)

(35)

Similarly



F120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890119894119904119906w (119906 119883119895 minus 119906) 119889119906

L120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890minus119904119906w (119906119883119895 minus 119906) 119889119906(37)


= 1119879119873119879120583 = 1119879119872119879

(38)




120573 = 1(1119872119879)sum119872119879119894=1 119884119894 (40)





120594 (119904) = [ + 120583 + 120575 minus L119891 (119904)] (119904 minus 120573) + 120583120573 (42)


1205941015840 (119904) = + 120583 + 120575 minus L119891 (119904) minus 119904L1198911015840 (119904)+ 120573L1198911015840 (119904)

= + 120583 + 120575 minus 1119873119879

119873119879sum119895=1

119890minus119904119883119895

+ 1119873119879

119873119879sum119895=1

119904119883119895119890minus119904119883119895

minus 120573 1119873119879

119873119879sum119895=1

119883119895119890minus119904119883119895

ge 120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895

119886119904997888997888rarr120583 minus 120582120573E [119883] gt 0

(43)





FΦ(119904) = F119867(119904)1 minus F119891120575 (119904) (44)

where

F119867(119904) = + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)

120588 + 119894119904 (F120596 (119904) minus L120596 (120588))]

F119891120575 (119904) = + 120583 + 120575 [F119891 (119904)

+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minus L119891(120588))]

(45)


Φ119870119886 (119906) fl 119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) 0 le 119906 le 119886

(46)





119867119895 (119909) = int1199090119906119895w (119906 119909 minus 119906) 119889119906 119895 = 0 1 2 (47)


intinfin0

119906119895120596 (119906) 119889119906 = E [119867119895 (119883)] (48)





10038171003817100381710038171003817Φ minus Φ119870119886


10038171003817100381710038171003817 (49)





1003817100381710038171003817Φ minus Φ1198701198861003817100381710038171003817 le 119862 119870 + 11198862119898+1 + 119886119870 minus 1 (50)


120588 minus 120588 = 119874119901 (119879minus12) (51)




where



(54)


119860119879 = 100381610038161003816100381610038161205941015840 (120588lowast)10038161003816100381610038161003816 gt 12 (120583 minus 120582120573E [119883]) (55)

Since 120583 minus 120573(1119873119879) sum119873119879119895=1119883119895


P (119860119879) ge P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 ge 12 (120583 minus 120582120573E [119883]))

= 1 minus P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 lt 12 (120583 minus 120582120573E [119883]))


119873119879sum119895=1

119883119895


(56)

Furthermore



(57)



sup119904isin[0119870120587119886]



F119867(119904) minusF119867(119904)= + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(59)


where

1198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904) 1198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[int1198831198950


0119890119894119904119906w (119906119883 minus 119906) 119889119906)] + ( + 120583 + 120575


= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

(61)

where

1198921119904 (119909) = int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906 119909 ge 0 (62)

For 1198921119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E (1198921119904 (119883))1003816100381610038161003816= sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816E(int1198830119890119894119904119906w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816

le 100381610038161003816100381610038161003816100381610038161003816E(int119883

0w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816 = 10038171003817100381710038171198670 (119883)10038171003817100381710038171198751 lt infin

(63)

then

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816(

+ 120583 + 120575 minus 120582120582 + 120583 + 120575)1003816100381610038161003816100381610038161003816100381610038161003816 sdot10038171003817100381710038171198670 (119883)10038171003817100381710038171199011

= 119874119901 (119879minus12)(64)



G119870119877 = 119892 119892 = Re (1198921119904) 119904 isin [0 119870120587119886 ] G119870119868 = 119892 119892 = Im (1198921119904) 119904 isin [0 119870120587119886 ] (65)

Then we have

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

+ sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1003816100381610038161003816119892 (119909)1003816100381610038161003816 le sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816int119909

0119890119894119904119906w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090w (119906 119909 minus 119906) 119889119906 = 1198670 (119909)

(67)


119909


= 10038161003816100381610038161003816100381610038161003816minus int119909



= 1198671 (119909) 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

(68)


119873 (120576G119870119877 1198712 (119875)) le 119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171198752 (69)


le int1205750




E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))


radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)


1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(78)





100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018










2 Preliminaries



119879119904ℎ (119910) = intinfin119910


119890minus119904119909ℎ (119909 + 119910) 119889119909119910 ge 0

(5)



Φ(119906) = int1199060Φ (119906 minus 119909) 119891120575 (119909) 119889119909 + 119867 (119906) (6)

where


119906w (119906 119909 minus 119906) 119891 (119909) 119889119909

(7)



Remark 1 Let

120594 (119904) = [120582 + 120583 + 120575 minus 120582L119891 (119904)] (119904 minus 120573) + 120583120573 119904 ge 0 (9)



1205941015840 (119904) = 120582 + 120583 + 120575 minus 120582L119891 (119904) minus 120582119904L1198911015840 (119904)+ 120582120573L1198911015840 (119904)

= 120582 + 120583 + 120575 minus 120582intinfin0

119890minus119904119909119891 (119909) 119889119909+ 120582intinfin


0119909119890minus119904119909119889119909

ge 120583 + 120575 minus 120582120573E [119883] gt 0

(10)




intinfin0

intinfin0

(1 + 119909)w (119909 119910) 119891 (119909 + 119910) 119889119910119889119909 lt infin (11)


w (119909 119910) le 119862 (1 + 119909)1205721 (1 + 119910)1205722 (12)


sup119906ge0

119879120588120596 (119906) le intinfin0

intinfin0

w (119909 119910) 119891 (119909 + 119910) 119889119910119889119909lt infin

(13)

thus

intinfin0

(119879120588120596 (119906))2 119889119906 le sup119906ge0

119879120588120596 (119906) times intinfin0

119879120588120596 (119906) 119889119906lt infin

(14)


intinfin0

(120596 (119906))2 119889119906 = sup119906ge0



0120596 (119906) 119889119906 lt infin

(15)







119890119894119904119906Φ(119906) 119889119906= intinfin

0119890119894119904119906 int119906

0Φ (119906 minus 119909) 119891120575 (119909) 119889119909119889119906

+ intinfin0

119890119894119904119906119867(119906) 119889119906= intinfin

0intinfin0

119890119894119904(119911+119909)Φ(119911) 119889119911119889119909 +F119867(119904)= FΦ (119904)F119891120575 (119904) +F119867(119904)

(16)

leading to

FΦ (119904) = F119867(119904)1 minusF119891120575 (119904) (17)


0119890119894119904119906119867(119906) 119889119906

= 120582120582 + 120583 + 120575 [intinfin0

119890119894119904119906120596 (119906) 119889119906+ (120573 minus 120588)intinfin

0119890119894119904119906119879120588120596 (119906) 119889119906]

= 120582120582 + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)intinfin

0119890119894119904119906119879120588120596 (119906) 119889119906]

(18)

where

intinfin0

119890119894119904119906119879120588120596 (119906) 119889119906 = intinfin0

119890119894119904119906 intinfin119906


0intinfin119906


0119890minus120588119910120596 (119910)int119910

0119890(119894119904+120588)119906119889119906119889119910

= 1120588 + 119894119904 (F120596 (119904) minusL120596 (120588))

(19)

Then we obtain

F119867(119904) = 120582120582 + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)120588 + 119894119904 (F120596 (119904) minusL120596 (120588))]

(20)


+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minusL119891 (120588))] (21)



ℎ (119909) = infinsum119896=0

1015840 21198862 minus 1198861 int1198862

1198861


(22)



ℎ119886 (119909) = ℎ (119909) sdot 1(0le119909le119886) 119886 gt 0 (23)


= infinsum119896=0

1015840 2119886 int1198860ℎ (119909) cos(119896120587119909119886)119889119909 cos (119896120587119909119886)

= infinsum119896=0

1015840 2119886 Reint1198860ℎ (119909) 119890119894(119896120587119886)119909119889119909 cos (119896120587119909119886)

0 le 119909 le 119886

(24)


Fℎ (119896120587119886 ) = int+infin0

ℎ (119909) 119890119894(119896120587119886)119909119889119909asymp int119886

0ℎ (119909) 119890119894(119896120587119886)119909119889119909

(25)

Thus


1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(26)



ℎ (119909) asymp 119870minus1sum119896=0

1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(27)


Φ (119906) asymp Φ119870119886 (119906)fl

119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) (28)







119880119895Δ 119895 = 0 1 2 119899 (29)


119873119895Δ 1198831 1198832 119883119873119895Δ 119895 = 1 119899 (30)


119872119895Δ 1198841 1198842 119884119873119895Δ 119895 = 1 119899 (31)




120582 120583 120588 120573 F119891 F120596 L119891(120588) L120596 (120588) (32)


F119891 (119904) = 1119873119879

119873119879sum119895=1

119890119894119904119883119895 (33)


L119891 (119904) = 1119873119879

119873119879sum119895=1

119890minus119904119883119895 (34)


0119890119894119904119906 intinfin

119906w (119906 119909 minus 119906) 119889119909119889119906

= intinfin0

int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909

= E(int1198830119890119894119904119906w (119906 119883 minus 119906) 119889119906)

(35)

Similarly



F120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890119894119904119906w (119906 119883119895 minus 119906) 119889119906

L120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890minus119904119906w (119906119883119895 minus 119906) 119889119906(37)


= 1119879119873119879120583 = 1119879119872119879

(38)




120573 = 1(1119872119879)sum119872119879119894=1 119884119894 (40)





120594 (119904) = [ + 120583 + 120575 minus L119891 (119904)] (119904 minus 120573) + 120583120573 (42)


1205941015840 (119904) = + 120583 + 120575 minus L119891 (119904) minus 119904L1198911015840 (119904)+ 120573L1198911015840 (119904)

= + 120583 + 120575 minus 1119873119879

119873119879sum119895=1

119890minus119904119883119895

+ 1119873119879

119873119879sum119895=1

119904119883119895119890minus119904119883119895

minus 120573 1119873119879

119873119879sum119895=1

119883119895119890minus119904119883119895

ge 120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895

119886119904997888997888rarr120583 minus 120582120573E [119883] gt 0

(43)





FΦ(119904) = F119867(119904)1 minus F119891120575 (119904) (44)

where

F119867(119904) = + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)

120588 + 119894119904 (F120596 (119904) minus L120596 (120588))]

F119891120575 (119904) = + 120583 + 120575 [F119891 (119904)

+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minus L119891(120588))]

(45)


Φ119870119886 (119906) fl 119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) 0 le 119906 le 119886

(46)





119867119895 (119909) = int1199090119906119895w (119906 119909 minus 119906) 119889119906 119895 = 0 1 2 (47)


intinfin0

119906119895120596 (119906) 119889119906 = E [119867119895 (119883)] (48)





10038171003817100381710038171003817Φ minus Φ119870119886


10038171003817100381710038171003817 (49)





1003817100381710038171003817Φ minus Φ1198701198861003817100381710038171003817 le 119862 119870 + 11198862119898+1 + 119886119870 minus 1 (50)


120588 minus 120588 = 119874119901 (119879minus12) (51)




where



(54)


119860119879 = 100381610038161003816100381610038161205941015840 (120588lowast)10038161003816100381610038161003816 gt 12 (120583 minus 120582120573E [119883]) (55)

Since 120583 minus 120573(1119873119879) sum119873119879119895=1119883119895


P (119860119879) ge P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 ge 12 (120583 minus 120582120573E [119883]))

= 1 minus P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 lt 12 (120583 minus 120582120573E [119883]))


119873119879sum119895=1

119883119895


(56)

Furthermore



(57)



sup119904isin[0119870120587119886]



F119867(119904) minusF119867(119904)= + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(59)


where

1198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904) 1198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[int1198831198950


0119890119894119904119906w (119906119883 minus 119906) 119889119906)] + ( + 120583 + 120575


= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

(61)

where

1198921119904 (119909) = int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906 119909 ge 0 (62)

For 1198921119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E (1198921119904 (119883))1003816100381610038161003816= sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816E(int1198830119890119894119904119906w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816

le 100381610038161003816100381610038161003816100381610038161003816E(int119883

0w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816 = 10038171003817100381710038171198670 (119883)10038171003817100381710038171198751 lt infin

(63)

then

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816(

+ 120583 + 120575 minus 120582120582 + 120583 + 120575)1003816100381610038161003816100381610038161003816100381610038161003816 sdot10038171003817100381710038171198670 (119883)10038171003817100381710038171199011

= 119874119901 (119879minus12)(64)



G119870119877 = 119892 119892 = Re (1198921119904) 119904 isin [0 119870120587119886 ] G119870119868 = 119892 119892 = Im (1198921119904) 119904 isin [0 119870120587119886 ] (65)

Then we have

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

+ sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1003816100381610038161003816119892 (119909)1003816100381610038161003816 le sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816int119909

0119890119894119904119906w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090w (119906 119909 minus 119906) 119889119906 = 1198670 (119909)

(67)


119909


= 10038161003816100381610038161003816100381610038161003816minus int119909



= 1198671 (119909) 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

(68)


119873 (120576G119870119877 1198712 (119875)) le 119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171198752 (69)


le int1205750




E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))


radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)


1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(78)





100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018











119890119894119904119906Φ(119906) 119889119906= intinfin

0119890119894119904119906 int119906

0Φ (119906 minus 119909) 119891120575 (119909) 119889119909119889119906

+ intinfin0

119890119894119904119906119867(119906) 119889119906= intinfin

0intinfin0

119890119894119904(119911+119909)Φ(119911) 119889119911119889119909 +F119867(119904)= FΦ (119904)F119891120575 (119904) +F119867(119904)

(16)

leading to

FΦ (119904) = F119867(119904)1 minusF119891120575 (119904) (17)


0119890119894119904119906119867(119906) 119889119906

= 120582120582 + 120583 + 120575 [intinfin0

119890119894119904119906120596 (119906) 119889119906+ (120573 minus 120588)intinfin

0119890119894119904119906119879120588120596 (119906) 119889119906]

= 120582120582 + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)intinfin

0119890119894119904119906119879120588120596 (119906) 119889119906]

(18)

where

intinfin0

119890119894119904119906119879120588120596 (119906) 119889119906 = intinfin0

119890119894119904119906 intinfin119906


0intinfin119906


0119890minus120588119910120596 (119910)int119910

0119890(119894119904+120588)119906119889119906119889119910

= 1120588 + 119894119904 (F120596 (119904) minusL120596 (120588))

(19)

Then we obtain

F119867(119904) = 120582120582 + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)120588 + 119894119904 (F120596 (119904) minusL120596 (120588))]

(20)


+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minusL119891 (120588))] (21)



ℎ (119909) = infinsum119896=0

1015840 21198862 minus 1198861 int1198862

1198861


(22)



ℎ119886 (119909) = ℎ (119909) sdot 1(0le119909le119886) 119886 gt 0 (23)


= infinsum119896=0

1015840 2119886 int1198860ℎ (119909) cos(119896120587119909119886)119889119909 cos (119896120587119909119886)

= infinsum119896=0

1015840 2119886 Reint1198860ℎ (119909) 119890119894(119896120587119886)119909119889119909 cos (119896120587119909119886)

0 le 119909 le 119886

(24)


Fℎ (119896120587119886 ) = int+infin0

ℎ (119909) 119890119894(119896120587119886)119909119889119909asymp int119886

0ℎ (119909) 119890119894(119896120587119886)119909119889119909

(25)

Thus


1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(26)



ℎ (119909) asymp 119870minus1sum119896=0

1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(27)


Φ (119906) asymp Φ119870119886 (119906)fl

119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) (28)







119880119895Δ 119895 = 0 1 2 119899 (29)


119873119895Δ 1198831 1198832 119883119873119895Δ 119895 = 1 119899 (30)


119872119895Δ 1198841 1198842 119884119873119895Δ 119895 = 1 119899 (31)




120582 120583 120588 120573 F119891 F120596 L119891(120588) L120596 (120588) (32)


F119891 (119904) = 1119873119879

119873119879sum119895=1

119890119894119904119883119895 (33)


L119891 (119904) = 1119873119879

119873119879sum119895=1

119890minus119904119883119895 (34)


0119890119894119904119906 intinfin

119906w (119906 119909 minus 119906) 119889119909119889119906

= intinfin0

int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909

= E(int1198830119890119894119904119906w (119906 119883 minus 119906) 119889119906)

(35)

Similarly



F120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890119894119904119906w (119906 119883119895 minus 119906) 119889119906

L120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890minus119904119906w (119906119883119895 minus 119906) 119889119906(37)


= 1119879119873119879120583 = 1119879119872119879

(38)




120573 = 1(1119872119879)sum119872119879119894=1 119884119894 (40)





120594 (119904) = [ + 120583 + 120575 minus L119891 (119904)] (119904 minus 120573) + 120583120573 (42)


1205941015840 (119904) = + 120583 + 120575 minus L119891 (119904) minus 119904L1198911015840 (119904)+ 120573L1198911015840 (119904)

= + 120583 + 120575 minus 1119873119879

119873119879sum119895=1

119890minus119904119883119895

+ 1119873119879

119873119879sum119895=1

119904119883119895119890minus119904119883119895

minus 120573 1119873119879

119873119879sum119895=1

119883119895119890minus119904119883119895

ge 120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895

119886119904997888997888rarr120583 minus 120582120573E [119883] gt 0

(43)





FΦ(119904) = F119867(119904)1 minus F119891120575 (119904) (44)

where

F119867(119904) = + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)

120588 + 119894119904 (F120596 (119904) minus L120596 (120588))]

F119891120575 (119904) = + 120583 + 120575 [F119891 (119904)

+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minus L119891(120588))]

(45)


Φ119870119886 (119906) fl 119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) 0 le 119906 le 119886

(46)





119867119895 (119909) = int1199090119906119895w (119906 119909 minus 119906) 119889119906 119895 = 0 1 2 (47)


intinfin0

119906119895120596 (119906) 119889119906 = E [119867119895 (119883)] (48)





10038171003817100381710038171003817Φ minus Φ119870119886


10038171003817100381710038171003817 (49)





1003817100381710038171003817Φ minus Φ1198701198861003817100381710038171003817 le 119862 119870 + 11198862119898+1 + 119886119870 minus 1 (50)


120588 minus 120588 = 119874119901 (119879minus12) (51)




where



(54)


119860119879 = 100381610038161003816100381610038161205941015840 (120588lowast)10038161003816100381610038161003816 gt 12 (120583 minus 120582120573E [119883]) (55)

Since 120583 minus 120573(1119873119879) sum119873119879119895=1119883119895


P (119860119879) ge P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 ge 12 (120583 minus 120582120573E [119883]))

= 1 minus P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 lt 12 (120583 minus 120582120573E [119883]))


119873119879sum119895=1

119883119895


(56)

Furthermore



(57)



sup119904isin[0119870120587119886]



F119867(119904) minusF119867(119904)= + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(59)


where

1198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904) 1198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[int1198831198950


0119890119894119904119906w (119906119883 minus 119906) 119889119906)] + ( + 120583 + 120575


= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

(61)

where

1198921119904 (119909) = int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906 119909 ge 0 (62)

For 1198921119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E (1198921119904 (119883))1003816100381610038161003816= sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816E(int1198830119890119894119904119906w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816

le 100381610038161003816100381610038161003816100381610038161003816E(int119883

0w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816 = 10038171003817100381710038171198670 (119883)10038171003817100381710038171198751 lt infin

(63)

then

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816(

+ 120583 + 120575 minus 120582120582 + 120583 + 120575)1003816100381610038161003816100381610038161003816100381610038161003816 sdot10038171003817100381710038171198670 (119883)10038171003817100381710038171199011

= 119874119901 (119879minus12)(64)



G119870119877 = 119892 119892 = Re (1198921119904) 119904 isin [0 119870120587119886 ] G119870119868 = 119892 119892 = Im (1198921119904) 119904 isin [0 119870120587119886 ] (65)

Then we have

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

+ sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1003816100381610038161003816119892 (119909)1003816100381610038161003816 le sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816int119909

0119890119894119904119906w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090w (119906 119909 minus 119906) 119889119906 = 1198670 (119909)

(67)


119909


= 10038161003816100381610038161003816100381610038161003816minus int119909



= 1198671 (119909) 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

(68)


119873 (120576G119870119877 1198712 (119875)) le 119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171198752 (69)


le int1205750




E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))


radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)


1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(78)





100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018










ℎ (119909) asymp 119870minus1sum119896=0

1015840 2119886 ReFℎ (119896120587119886 ) cos (119896120587119909119886) 0 le 119909 le 119886

(27)


Φ (119906) asymp Φ119870119886 (119906)fl

119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) (28)







119880119895Δ 119895 = 0 1 2 119899 (29)


119873119895Δ 1198831 1198832 119883119873119895Δ 119895 = 1 119899 (30)


119872119895Δ 1198841 1198842 119884119873119895Δ 119895 = 1 119899 (31)




120582 120583 120588 120573 F119891 F120596 L119891(120588) L120596 (120588) (32)


F119891 (119904) = 1119873119879

119873119879sum119895=1

119890119894119904119883119895 (33)


L119891 (119904) = 1119873119879

119873119879sum119895=1

119890minus119904119883119895 (34)


0119890119894119904119906 intinfin

119906w (119906 119909 minus 119906) 119889119909119889119906

= intinfin0

int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909

= E(int1198830119890119894119904119906w (119906 119883 minus 119906) 119889119906)

(35)

Similarly



F120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890119894119904119906w (119906 119883119895 minus 119906) 119889119906

L120596 (119904) = 1119873119879

119873119879sum119895=1

int1198831198950

119890minus119904119906w (119906119883119895 minus 119906) 119889119906(37)


= 1119879119873119879120583 = 1119879119872119879

(38)




120573 = 1(1119872119879)sum119872119879119894=1 119884119894 (40)





120594 (119904) = [ + 120583 + 120575 minus L119891 (119904)] (119904 minus 120573) + 120583120573 (42)


1205941015840 (119904) = + 120583 + 120575 minus L119891 (119904) minus 119904L1198911015840 (119904)+ 120573L1198911015840 (119904)

= + 120583 + 120575 minus 1119873119879

119873119879sum119895=1

119890minus119904119883119895

+ 1119873119879

119873119879sum119895=1

119904119883119895119890minus119904119883119895

minus 120573 1119873119879

119873119879sum119895=1

119883119895119890minus119904119883119895

ge 120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895

119886119904997888997888rarr120583 minus 120582120573E [119883] gt 0

(43)





FΦ(119904) = F119867(119904)1 minus F119891120575 (119904) (44)

where

F119867(119904) = + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)

120588 + 119894119904 (F120596 (119904) minus L120596 (120588))]

F119891120575 (119904) = + 120583 + 120575 [F119891 (119904)

+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minus L119891(120588))]

(45)


Φ119870119886 (119906) fl 119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) 0 le 119906 le 119886

(46)





119867119895 (119909) = int1199090119906119895w (119906 119909 minus 119906) 119889119906 119895 = 0 1 2 (47)


intinfin0

119906119895120596 (119906) 119889119906 = E [119867119895 (119883)] (48)





10038171003817100381710038171003817Φ minus Φ119870119886


10038171003817100381710038171003817 (49)





1003817100381710038171003817Φ minus Φ1198701198861003817100381710038171003817 le 119862 119870 + 11198862119898+1 + 119886119870 minus 1 (50)


120588 minus 120588 = 119874119901 (119879minus12) (51)




where



(54)


119860119879 = 100381610038161003816100381610038161205941015840 (120588lowast)10038161003816100381610038161003816 gt 12 (120583 minus 120582120573E [119883]) (55)

Since 120583 minus 120573(1119873119879) sum119873119879119895=1119883119895


P (119860119879) ge P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 ge 12 (120583 minus 120582120573E [119883]))

= 1 minus P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 lt 12 (120583 minus 120582120573E [119883]))


119873119879sum119895=1

119883119895


(56)

Furthermore



(57)



sup119904isin[0119870120587119886]



F119867(119904) minusF119867(119904)= + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(59)


where

1198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904) 1198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[int1198831198950


0119890119894119904119906w (119906119883 minus 119906) 119889119906)] + ( + 120583 + 120575


= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

(61)

where

1198921119904 (119909) = int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906 119909 ge 0 (62)

For 1198921119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E (1198921119904 (119883))1003816100381610038161003816= sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816E(int1198830119890119894119904119906w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816

le 100381610038161003816100381610038161003816100381610038161003816E(int119883

0w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816 = 10038171003817100381710038171198670 (119883)10038171003817100381710038171198751 lt infin

(63)

then

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816(

+ 120583 + 120575 minus 120582120582 + 120583 + 120575)1003816100381610038161003816100381610038161003816100381610038161003816 sdot10038171003817100381710038171198670 (119883)10038171003817100381710038171199011

= 119874119901 (119879minus12)(64)



G119870119877 = 119892 119892 = Re (1198921119904) 119904 isin [0 119870120587119886 ] G119870119868 = 119892 119892 = Im (1198921119904) 119904 isin [0 119870120587119886 ] (65)

Then we have

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

+ sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1003816100381610038161003816119892 (119909)1003816100381610038161003816 le sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816int119909

0119890119894119904119906w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090w (119906 119909 minus 119906) 119889119906 = 1198670 (119909)

(67)


119909


= 10038161003816100381610038161003816100381610038161003816minus int119909



= 1198671 (119909) 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

(68)


119873 (120576G119870119877 1198712 (119875)) le 119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171198752 (69)


le int1205750




E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))


radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)


1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(78)





100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018












120594 (119904) = [ + 120583 + 120575 minus L119891 (119904)] (119904 minus 120573) + 120583120573 (42)


1205941015840 (119904) = + 120583 + 120575 minus L119891 (119904) minus 119904L1198911015840 (119904)+ 120573L1198911015840 (119904)

= + 120583 + 120575 minus 1119873119879

119873119879sum119895=1

119890minus119904119883119895

+ 1119873119879

119873119879sum119895=1

119904119883119895119890minus119904119883119895

minus 120573 1119873119879

119873119879sum119895=1

119883119895119890minus119904119883119895

ge 120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895

119886119904997888997888rarr120583 minus 120582120573E [119883] gt 0

(43)





FΦ(119904) = F119867(119904)1 minus F119891120575 (119904) (44)

where

F119867(119904) = + 120583 + 120575 [F120596 (119904)+ (120573 minus 120588)

120588 + 119894119904 (F120596 (119904) minus L120596 (120588))]

F119891120575 (119904) = + 120583 + 120575 [F119891 (119904)

+ (120573 minus 120588)120588 + 119894119904 (F119891 (119904) minus L119891(120588))]

(45)


Φ119870119886 (119906) fl 119870minus1sum119896=0

1015840 2119886 ReFΦ(119896120587119886 ) cos (119896120587119909119886) 0 le 119906 le 119886

(46)





119867119895 (119909) = int1199090119906119895w (119906 119909 minus 119906) 119889119906 119895 = 0 1 2 (47)


intinfin0

119906119895120596 (119906) 119889119906 = E [119867119895 (119883)] (48)





10038171003817100381710038171003817Φ minus Φ119870119886


10038171003817100381710038171003817 (49)





1003817100381710038171003817Φ minus Φ1198701198861003817100381710038171003817 le 119862 119870 + 11198862119898+1 + 119886119870 minus 1 (50)


120588 minus 120588 = 119874119901 (119879minus12) (51)




where



(54)


119860119879 = 100381610038161003816100381610038161205941015840 (120588lowast)10038161003816100381610038161003816 gt 12 (120583 minus 120582120573E [119883]) (55)

Since 120583 minus 120573(1119873119879) sum119873119879119895=1119883119895


P (119860119879) ge P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 ge 12 (120583 minus 120582120573E [119883]))

= 1 minus P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 lt 12 (120583 minus 120582120573E [119883]))


119873119879sum119895=1

119883119895


(56)

Furthermore



(57)



sup119904isin[0119870120587119886]



F119867(119904) minusF119867(119904)= + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(59)


where

1198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904) 1198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[int1198831198950


0119890119894119904119906w (119906119883 minus 119906) 119889119906)] + ( + 120583 + 120575


= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

(61)

where

1198921119904 (119909) = int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906 119909 ge 0 (62)

For 1198921119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E (1198921119904 (119883))1003816100381610038161003816= sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816E(int1198830119890119894119904119906w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816

le 100381610038161003816100381610038161003816100381610038161003816E(int119883

0w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816 = 10038171003817100381710038171198670 (119883)10038171003817100381710038171198751 lt infin

(63)

then

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816(

+ 120583 + 120575 minus 120582120582 + 120583 + 120575)1003816100381610038161003816100381610038161003816100381610038161003816 sdot10038171003817100381710038171198670 (119883)10038171003817100381710038171199011

= 119874119901 (119879minus12)(64)



G119870119877 = 119892 119892 = Re (1198921119904) 119904 isin [0 119870120587119886 ] G119870119868 = 119892 119892 = Im (1198921119904) 119904 isin [0 119870120587119886 ] (65)

Then we have

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

+ sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1003816100381610038161003816119892 (119909)1003816100381610038161003816 le sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816int119909

0119890119894119904119906w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090w (119906 119909 minus 119906) 119889119906 = 1198670 (119909)

(67)


119909


= 10038161003816100381610038161003816100381610038161003816minus int119909



= 1198671 (119909) 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

(68)


119873 (120576G119870119877 1198712 (119875)) le 119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171198752 (69)


le int1205750




E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))


radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)


1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(78)





100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018











10038171003817100381710038171003817Φ minus Φ119870119886


10038171003817100381710038171003817 (49)





1003817100381710038171003817Φ minus Φ1198701198861003817100381710038171003817 le 119862 119870 + 11198862119898+1 + 119886119870 minus 1 (50)


120588 minus 120588 = 119874119901 (119879minus12) (51)




where



(54)


119860119879 = 100381610038161003816100381610038161205941015840 (120588lowast)10038161003816100381610038161003816 gt 12 (120583 minus 120582120573E [119883]) (55)

Since 120583 minus 120573(1119873119879) sum119873119879119895=1119883119895


P (119860119879) ge P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 ge 12 (120583 minus 120582120573E [119883]))

= 1 minus P(120583 minus 120573 1119873119879

119873119879sum119895=1

119883119895 lt 12 (120583 minus 120582120573E [119883]))


119873119879sum119895=1

119883119895


(56)

Furthermore



(57)



sup119904isin[0119870120587119886]



F119867(119904) minusF119867(119904)= + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(59)


where

1198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904) 1198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[int1198831198950


0119890119894119904119906w (119906119883 minus 119906) 119889119906)] + ( + 120583 + 120575


= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

(61)

where

1198921119904 (119909) = int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906 119909 ge 0 (62)

For 1198921119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E (1198921119904 (119883))1003816100381610038161003816= sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816E(int1198830119890119894119904119906w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816

le 100381610038161003816100381610038161003816100381610038161003816E(int119883

0w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816 = 10038171003817100381710038171198670 (119883)10038171003817100381710038171198751 lt infin

(63)

then

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816(

+ 120583 + 120575 minus 120582120582 + 120583 + 120575)1003816100381610038161003816100381610038161003816100381610038161003816 sdot10038171003817100381710038171198670 (119883)10038171003817100381710038171199011

= 119874119901 (119879minus12)(64)



G119870119877 = 119892 119892 = Re (1198921119904) 119904 isin [0 119870120587119886 ] G119870119868 = 119892 119892 = Im (1198921119904) 119904 isin [0 119870120587119886 ] (65)

Then we have

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

+ sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1003816100381610038161003816119892 (119909)1003816100381610038161003816 le sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816int119909

0119890119894119904119906w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090w (119906 119909 minus 119906) 119889119906 = 1198670 (119909)

(67)


119909


= 10038161003816100381610038161003816100381610038161003816minus int119909



= 1198671 (119909) 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

(68)


119873 (120576G119870119877 1198712 (119875)) le 119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171198752 (69)


le int1205750




E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))


radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)


1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(78)





100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018









where

1198681 = + 120583 + 120575F120596 (119904) minus 120582120582 + 120583 + 120575F120596 (119904) 1198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[int1198831198950


0119890119894119904119906w (119906119883 minus 119906) 119889119906)] + ( + 120583 + 120575


= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

(61)

where

1198921119904 (119909) = int1199090119890119894119904119906w (119906 119909 minus 119906) 119889119906 119909 ge 0 (62)

For 1198921119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E (1198921119904 (119883))1003816100381610038161003816= sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816E(int1198830119890119894119904119906w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816

le 100381610038161003816100381610038161003816100381610038161003816E(int119883

0w (119906119883 minus 119906) 119889119906)100381610038161003816100381610038161003816100381610038161003816 = 10038171003817100381710038171198670 (119883)10038171003817100381710038171198751 lt infin

(63)

then

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E (1198921119904 (119883))

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816(

+ 120583 + 120575 minus 120582120582 + 120583 + 120575)1003816100381610038161003816100381610038161003816100381610038161003816 sdot10038171003817100381710038171198670 (119883)10038171003817100381710038171199011

= 119874119901 (119879minus12)(64)



G119870119877 = 119892 119892 = Re (1198921119904) 119904 isin [0 119870120587119886 ] G119870119868 = 119892 119892 = Im (1198921119904) 119904 isin [0 119870120587119886 ] (65)

Then we have

sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

+ sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

(66)



1003816100381610038161003816119892 (119909)1003816100381610038161003816 le sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816int119909

0119890119894119904119906w (119906 119909 minus 119906) 11988911990610038161003816100381610038161003816100381610038161003816

le int1199090w (119906 119909 minus 119906) 119889119906 = 1198670 (119909)

(67)


119909


= 10038161003816100381610038161003816100381610038161003816minus int119909



= 1198671 (119909) 10038161003816100381610038161199041 minus 11990421003816100381610038161003816

(68)


119873 (120576G119870119877 1198712 (119875)) le 119862119870120587120576119886 10038171003817100381710038171198671 (119909)10038171003817100381710038171198752 (69)


le int1205750




E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))


radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)


1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(78)





100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018










E( 1radic119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879)le 119869 (120575G119870119877 1198712 (119875))le int120575

0


then

E( 1119873119879

sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816)

= E( 1radic119873119879

E( 1radic119873119879

sdot sup119892isinG119870119877

10038161003816100381610038161003816100381610038161003816100381610038161003816119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816 | 119873119879))


radicE [119873119879] = radic log (119870119886)120582119879

(72)

Therefore

sup119892isinG119870119877

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(73)

Similarly

sup119892isinG119870119868

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[119892 (119883119895) minus E (119892 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(74)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198921119904 (119883119895) minus E (1198921119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (75)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (76)


1198682 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F120596 (119904) minus L120596 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(77)

where

1198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


1198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



(78)





100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018












100381610038161003816100381611989711003816100381610038161003816 le + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950


119873119879sum119895=1

1198671 (119883119895)

(80)


100381610038161003816100381611989711003816100381610038161003816 = 119874119901 (119879minus12) (81)

For 1198972 we have1198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

int1198831198950



= (120573 minus 120588) + 120583 + 120575

120588 + 119894119904120588 + 119894119904 1119873119879

sdot 119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))] + (120588 + 119894119904)


sdot E (1198922119904 (119883))

(82)

where

1198922119904 (119909) = int1199090



(83)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198922119904 (119883119895) minus E (1198922119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (84)

In addition we have


= 119874119901 (119879minus12)(85)


119904isin[0119870120587119886]

1003816100381610038161003816E [1198922119904 (119883)]1003816100381610038161003816le intinfin


le intinfin0

int1199090119906w (119906 119909 minus 119906) 119889119906119891 (119909) 119889119909 = 10038171003817100381710038171198671 (119883)10038171003817100381710038171198751

lt infin

(86)

thus

sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


sdot E (1198922119904 (119883))10038161003816100381610038161003816100381610038161003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816100381610038161003816(120588 + 119894119904)


10038161003816100381610038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171198671 (119883)10038171003817100381710038171199011 = 119874119901 (119879minus12)

(87)


sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (88)


sup119904isin[0119870120587119886]

100381610038161003816100381611986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 ) (89)




119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018











119904isin[0119870120587119886]



F119891120575 (119904) minusF119891120575 (119904)= + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904)

+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891(120588)]


(91)

where

1198681198681 = + 120583 + 120575F119891 (119904) minus 120582120582 + 120583 + 120575F119891 (119904) 1198681198682 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]



= + 120583 + 120575 1119873119879

119873119879sum119895=1

[119890119894119904119883119895 minus E [119890119894119904119883]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [119890119894119904119883]

= + 120583 + 120575 1119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E [1198923119904 (119883)]]+ ( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

(93)

where

1198923119904 (119909) = 119890119894119904119909 119909 ge 0 (94)

For 1198923119904(119909) we havesup

119904isin[0119870120587119886]

1003816100381610038161003816E [1198923119904 (119883)]1003816100381610038161003816 = sup119904isin[0119870120587119886]

10038161003816100381610038161003816100381610038161003816intinfin

0119890119894119904119909119891 (119909) 11988911990910038161003816100381610038161003816100381610038161003816

le 1 lt infin(95)

leading to

sup119904isin[0119870120587119886]

1003816100381610038161003816100381610038161003816100381610038161003816( + 120583 + 120575 minus 120582120582 + 120583 + 120575)E [1198923119904 (119883)]

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

+ 120583 + 120575 minus 120582120582 + 120583 + 1205751003816100381610038161003816100381610038161003816100381610038161003816 = 119874119901 (119879minus12)

(96)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198923119904 (119883119895) minus E (1198923119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (97)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986811003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (98)


(120573 minus 120588)120588 + 119894119904 [F119891 (119904) minus L119891 (120588)]


(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


= + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


+ + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(99)


where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018









where

1198971198971 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


1198971198972 = + 120583 + 120575(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


(100)


100381610038161003816100381611989711989711003816100381610038161003816 = 119874119901 (119879minus12) (101)

For 1198971198972 we have1198971198972 = + 120583 + 120575

(120573 minus 120588)120588 + 119894119904 1119873119879

119873119879sum119895=1


119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]


sdot E (1198924119904 (119883))

(102)

where


119890119894119904(119909minus119910)minus120588119910119889119910 119909 ge 0 (103)


sup119904isin[0119870120587119886]

100381610038161003816100381610038161003816100381610038161003816100381610038161119873119879

119873119879sum119895=1

[1198924119904 (119883119895) minus E (1198924119904 (119883))]10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119874119901(radic log (119870119886)119879 ) (104)


sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 = 119874119901(radic log (119870119886)119879 ) (105)


sup119904isin[0119870120587119886]

100381610038161003816100381611986811986821003816100381610038161003816 le sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816 + sup119904isin[0119870120587119886]

100381610038161003816100381611989711989721003816100381610038161003816= 119874119901(radic log (119870119886)119879 )

(106)




100381710038171003817100381710038172 = 119874119901 ((119870119886) log (119870119886)119879 ) (107)

Proof First we have

10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 = int1198860


41198862 (ReFΦ(119896120587119886 ) minus FΦ(119896120587119886 ))2

sdot int1198860(cos(119896120587119886 119906))2 119889119906 = 2119886


10038161003816100381610038161003816100381610038161003816FΦ(119896120587119886 ) minus FΦ(119896120587119886 )100381610038161003816100381610038161003816100381610038162 le 2119870119886

sdot sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

(108)


sup119904isin[0119870120587119886]


1003816100381610038161003816100381610038161003816100381610038161003816F119867(119904)1 minusF119891120575 (119904) minus

F119867(119904)1 minus F119891120575 (119904)1003816100381610038161003816100381610038161003816100381610038161003816

= sup119904isin[0119870120587119886]


10038161003816100381610038161003816100381610038161003816100381610038161003816 (109)


(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018









(a1) (a2) (a3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



sup119904isin[0119870120587119886]



10038171003817100381710038171003817Φ119870119886 minus Φ119870119886

100381710038171003817100381710038172 le 2119870119886 sup119904isin[0119870120587119886]

10038161003816100381610038161003816FΦ (119904) minus FΦ (119904)100381610038161003816100381610038162

= 119874119901 ((119870119886) log (119870119886)119879 ) (111)



10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874(119870 + 11198862119898+1 ) + 119874( 119886119870 minus 1)+ 119874119901 ((119870119886) log (119870119886)119879 ) (112)


10038171003817100381710038171003817Φ minus Φ119870119886

100381710038171003817100381710038172 = 119874 (119870minus119898(119898+1))+ 119874119901 (119870119898(119898+1)log (119870)119879 ) (113)





1300300sum119895=1

Φ119870119886119895 (119906) 1300

300sum119895=1

1003816100381610038161003816100381610038161003816100381610038161003816Φ119870119886119895 (119906)Φ (119906) minus 11003816100381610038161003816100381610038161003816100381610038161003816

(114)


1300300sum119895=1

intinfin0

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906

asymp 1300300sum119895=1

int300

(Φ119870119886119895 (119906) minus Φ (119906))2 119889119906(115)




(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018









(b1) (b2) (b3)

minus010

0102030405060708

minus010

01020304050607

minus010

01020304050607

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156

T=260T=364

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0

01

02

03

04

05

06

0

01

02

03

04

05

06

0

01

02

03

04

05

06

5 10 15 20 25 300u

(c1) (c2) (c3)

5 10 15 20 25 300u

5 10 15 20 25 300u


T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

T=52T=156T=260

T=364true value

0010203040506

0010203040506

0010203040506

(d1) (d2) (d3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u






T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018









T=52T=156

T=260T=364

T=52T=156

T=260T=364

T=52T=156

T=260T=364

05

1015202530354045

0

5

10

15

20

25

0102030405060

(e1) (e2) (e3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


0102030405060708090

100

T=52T=156

T=260T=364

(f1)

0

50

100

150

T=52T=156

T=260T=364

(f2)

010203040506070

T=52T=156

T=260T=364

(f3)

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Fourier-CosineFFT

(g1)

Fourier-CosineFFT

(g2)

Fourier-CosineFFT

(g3)

010203040506070

01020304050607080

010203040506070

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u



Data Availability







Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018











Fouroer-Cosine

52 00242 00160 00378 00164 00130 00090156 00081 00056 00109 00056 00031 00038260 00047 00030 00064 00028 00023 00017364 00033 00021 00046 00021 00015 00014

FFT

52 00209 00171 00332 00169 00114 00090156 00086 00065 00117 00068 00052 00042260 00059 00051 00079 00049 00038 00031364 00048 00043 00054 00039 00032 00024

Fourier-CosineFFT

(h1)

Fourier-CosineFFT

(h2)

Fourier-CosineFFT

(h3)

0100200300400500600700800

0200400600800

10001200140016001800

0

500

1000

1500

2000

2500

5 10 15 20 25 300u

5 10 15 20 25 300u

5 10 15 20 25 300u


Acknowledgments


References







































































Journal of












Journal of







Volume 2018






Volume 2018




































































Journal of












Journal of







Volume 2018






Volume 2018































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Estimating the Gerber-Shiu Function in a Compound …downloads.hindawi.com/journals/ddns/2019/5071268.pdfGerber-Shiu function is proposed by the observed sample of the surplus process