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International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
On the Construction of Kumaraswamy-Epsilon
Distribution with Applications
Gongsin Isaac Esbond1 Saporu W O Funmilayo
2
1Department of Mathematical Sciences University of Maiduguri PMB 1069 Bama Road Maiduguri Borno State Nigeria
2National Mathematical Centre Abuja-Lokoja Expressway Kwali-Sheda P M B 118 Abuja Nigeria
Abstract A new probability distribution function of the Kumaraswamy-G family is introduced in this study It is constructed from a
parent distribution called the epsilon distributionThis new distribution has more flexible probability density function shapes that suggest
its possible use for example in modeling lifetime data generation processes Its hazard rate functions and order
statisticsdistributionswere derived The distribution significantly provided good fit to two real life datasets with efficient parameter
estimates within twice their standard errors These attest to its practical applicability
Keywords Kumaraswamy-G epsilon distribution hazard rate function order statistic deuterium excesses monthly tax revenue
1 Introduction
A beta-type distribution was introduced in 1980 for a double
bounded random variable [16] andnamed after the founder
Kumaraswamy A random variable 119883 is said to be
distributed according to the Kumaraswamy distribution with
parameters 119886 and 119887 if its density and cumulative distribution
functions are given respectively by
where in each case 0 lt 119909 lt 1 and119886 119887 gt 0The
distribution has application in hydrology reliability studies
[17] and other areas
Following the works of Jones [14] and Eugene et al [12]
Cordeiro and de Castro [6] proposed the use of
Kumaraswamy distribution as a generator of probability
distributions of continuous random variables with parent
distribution 119866 119909 These are referred to as the
Kumaraswamy-G distribution Its cumulative distribution
function is given by
where 119892 119909 =119889
119889119909119866 119909 The parameters 119886 and 119887 in the
equation (4) determine the skewness and tail weight
respectively
In this study a new distribution based on the Kumaraswamy
generator is introducedIt is called the Kumaraswamy-
epsilon distribution and subsequently denoted by the K-
epsilon distribution
2 Literature Review
Of recent interest in deriving new generalized probability
distributions has grown tremendously It has resulted in
generalized classes of univariate distributions after
introducing additional shape parameter to the baseline or
parent distributions The new generated families extend the
well-known classical probability distributions and at the
same time provide great flexibility in modeling data
generating processes For examples beta generator [12]
gamma generator [19] the Weibull-G family [3]
exponentiated family and generalized exponentiated family
[8] of density functions have all been introduced Many
probability distributions have been constructed using these
generators
Many members of the Kumaraswamy generator have been
constructed and applied For instance the Kumaraswamy
Gumbel [7] Kumaraswamy Laplace [1] Kumaraswamy
Pareto [4] Kumaraswamy generalized Rayleigh [13] the
Kumaraswamy Lindley distribution [5] Kumaraswamy
generalized gamma distribution [10] have been used in
modeling bathtub shaped hazard rate functions Also they
have the ability to model monotone and non-monotone
failure rate functions as such it can be useful in both
lifetime data analysis and reliability studies Oguntunde et al
[18] introduced the Kumaraswamy inverse exponential
distribution and studied some of its properties The
Kumaraswamy Weibull distribution was introduced in [9]
and used in the study of failure data
3 The K-epsilon distribution and its properties
The epsilon distribution was introduced in [11] and shown to
be asymptotically equivalent to the exponential distribution
For a random variable 119883 distributed according to the epsilon
distribution its probability density and cumulative
distribution functions are given respectively by
where 0 lt 119909 lt 120575 120575 120582 gt 0
Putting (6) into (3) and (4) the cumulative distribution and
probability density functions of the K-epsilon distribution
are given respectively by
Paper ID ART20202623 1021275ART20202623 1199
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
whereΥ = 1205752
1205752minus1199092 Ζ = 1 minus 119909+120575
120575minus119909 minus120582
2120575
0 lt 119909 lt 120575119886 119887 120575 120582 gt
0
The skewness and tail weight of the distribution are
controlled by the parameters 119886 and 119887 The plots of the K-
epsilon density function (8) at various parameter values are
given in Fig 1 below
Figure 1 K-Epsilon probability density plots at various parameter values
From (7) the quantile function of the K-epsilon distribution
is given by
where 0 lt 119901 lt 1 119886 119887 120575 120582 gt 0
Proposition 1 The K-epsilon distribution is a true probability density
function with unimodal shape property That is we want to
show that
(i) 119891119883 119909 is unimodal and
(ii) 119891119883 119909 120575
0119889119909 = 1
The proof of this proposition is given in the appendix
31 Moments of the K-epsilon distribution
The 119903119905119893 119903 = 1 2hellip moment of the K-epsilon can be
derived from the expression
From (10) it is obvious that the moments of the K-epsilon
distribution cannot be obtained in explicit form but can be
obtained by numerical integration for the desired moment(s)
when the parameter values are specified
32 Survival and hazard rate functions
Survival and hazard rate functions form the focal points in
survival analysis and reliability theory The survival
function which is the complementary cumulative
distribution or reliability function is defined as the
probability of surviving to the age of 119909 or exceeding age 119909
On the other hand the hazard rate function also known as
instantaneous failure rate or force of mortality is interpreted
as the amount of risk of failure associated with a unit of age
119909 The two functions are related however the hazard rate
a = 2 b = 04 120575 120582 = 4 (black) 25 (red)
15 (blue) 05 (green)
0 1 2 3 4 5 6X values
00
02
04
06
08
Den
sity
a = 2 120582 = 3 120575 = 6
b = 015 (black) 025 (red)
035 (blue) 05 (green)
0 1 2 3 4 5 6X values
00
02
04
06
08
Den
sity
a = 2 b = 04 120582 120575 = 6 (black) 10 (red)
15 (blue) 25 (green)
0 1 2 3 4 5 6X
values
00
02
04
06
08
Den
sity
0 1 2 3 4 5 6X values
00
02
04
06
08
Den
sity
b = 04 120582 = 3 120575 = 6
a = 2 (black) 3 (red)
5 (blue) 10 (green)
Paper ID ART20202623 1021275ART20202623 1200
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
function is more informative about the underlying
mechanism of failure
The survival function of the K-epsilon distribution is given
by
and its hazard rate function is given by
The plots of the hazard rate function of the K-epsilon
distribution at various parameter values are presented in Fig
2 below The shapes suggest that the K-epsilon hazard rate
function could be a good choice model for somefailure rate
datasets
Figure 2 K-Epsilon distribution hazard rate function at various parameter values
33 Order statistics distribution
Order statistics play important role in life testing and
reliability analysis Given a random sample 1199091 1199092 hellip 119909119899 of
a random variable 119883 having the K-epsilon distribution and
density functions 119865119883 and 119891119883 respectively The distribution of
the 119903119905119893order statistic 119883(119903) of the ordered sample 119909(1) hellip
119909(119903) hellip 119909(119899) is given by
119891119883 119903 119909 = 119886119887120582ΛΥ 1 minus Ζ Ζ119886minus1Ψ119887 119899minus119903+1 minus1 1 minusΨ119887 119903minus1(13)
Where Λ =119899
119903minus1 119899minus119903 Υ =
1205752
1205752minus1199092
Ζ = 1 minus 119909+120575
120575minus119909 minus120582
2120575
andΨ = 1minus Ζ119886 From (13) the K-epsilon
distribution of the first order statistic 119883(1) and the 119899119905119893 order
statistic 119883(119899) are given respectively by
4 Parameter Estimation
As noted earlier the explicit expressions for the K-epsilon
probability density cumulative distribution and quantile
functions has made the distribution a good candidate for the
deployment of the fitdistrplus package in R This has the
advantage of producing parameter estimates their standard
errors and many other important statistical properties
5 Application
The K-epsilon distribution is applied to 2 datasets
namelydeuterium excesses and tax revenue datasets These
are represented in boxplots in Fig 3 below
Paper ID ART20202623 1021275ART20202623 1201
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 3 Boxplots of Deuterium excesses and Monthly Tax
Revenue
The first dataset is hourly record of deuterium (heavy
hydrogen ndash one of the two stable isotopes of hydrogen)
excesses from 2015-11-11T0000 to 2015-11-24T2300 [2]
given in the appendix
Parameter estimates and Kolmogorov-Smirnov goodness-of-
fit test results are given in Table 1 while the fitted K-epsilon
densityplot is presented in Fig 3
Table 1 K-epsilon distribution fit results for deuterium excesses dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark
K-epsilon 15866 (01061) 22124 (959727) 00050 (00007) 97456 (02744) 67668 00518 (00781) Good fit
119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value
From Table 1 it could be observed that the K-epsilon
distribution has fitted the deuterium excesses datasetThe
results of the goodness-of-fit test shown in Table 1 above
provide sufficient evidence that the K-epsilon distribution
fits the dataset Also the standard errors of the parameter
estimates obtained indicate they are of good precision
Figure 3 K-Epsilon distribution fit of hydrogen
Isotope (Deuterium) excesses
The second dataset for application is the monthly records of
tax revenue of Egypt between January 2006 and November
2010 in 1000 million Egyptian pounds reported in [15] It is
presented in the appendix
The results for parameter estimate and goodness-of-fit test
are presented in Table 2 while the fitted K-epsilon
probability density plot is presented in Fig 4
Table 2 K-epsilon distribution fit results for Monthly Tax Revenue dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark
K-epsilon 638712 (18779) 01877 (00378) 06924 (00637) 20488 (02086) 187973 00735 (01771) Good fit
119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value
Table 2 also suggest that the K-epsilon distribution has
successfully fitted the tax revenue dataset The parameter
estimates are of good precision
Den
sity
00
00
05
01
00
15
Deuterium excesses0 2 4 6 8 10
Paper ID ART20202623 1021275ART20202623 1202
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 5 K-epsilon distribution fit of Monthly Tax Revenue
6 Conclusion
A new distribution called the Kumaraswamy epsilon (K-
epsilon) distribution has been derived in this study It is
constructed from the Kumaraswamy generator
(Kumaraswamy-G) of probability density functions Here
the parent distribution is the epsilon distribution The new
distribution unlike the beta distribution has explicit
expressions for the cumulative distribution and quantile
functions This makes parameter estimates much easier
particularly in R Although the moments of the K-epsilon
distribution could not be expressed in explicit form other
properties of the K-epsilon distribution such as the
distribution of order statistics survival and hazard rate
functions have been derived The shapes of the plots of the
probability density and hazard rate functions at various
parameter values suggest that the K-epsilon distribution
holds the potential for wide applications within its parameter
space
The K-epsilon distribution was applied to two different
lifetime datasets namely deuterium (hydrogen isotope)
excesses and monthly tax revenue The results of the fit of
this distribution to these datasets are good confirming its
wide usefulness in practical applications
References
[1] Aryal G (2015) Kumaraswamy Laplace Distribution
and its Applications Paper presented at the Seventh
International Conference on Dynamic Systems and
Applications and Fifth International Conference on
Neural Parallel and Scientific Computations
[2] Bonne J-L Werner M Meyer H Kipfstuhl S
Rabe B Behrens Melanie K SchAtildeparanicke L amp
Steen-Larsen H-C (2019) Water vapour isotopes
analyser calibrated data from POLARSTERN during
dockyard 2015-08 at Port of TromsAtildecedil Norway
PANGAEA httpsdoiorg101594PANGAEA897410
[3] Bourguignon M Silva R B amp Cordeiro G M (2014)
The Weibull-G family of probability distributions
Journal of Data Science 12 53 ndash 68
[4] Bourguignon M Silva R B Zea L M amp Cordeiro
G M (2013) The Kumaraswamy Pareto distribution
Journal of Statistical Theory and Applications 12 129-
144
[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new
customer lifetime duration distribution the
Kumaraswamy Lindley distribution International
Journal of Trade Economics and Finance Vol 5 No
5 441 ndash 444
[6] Cordeiro G M amp de Castro M (2011) A new family
of generalized distributions Journal of statistical
computation and simulation 81 883-898
httpdxdoiorg10108000949650903530745
[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)
The Kumaraswamy Gumbel distribution Statistical
Methods and Applications 21 139-168
[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27
[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)
The Kumaraswamy Weibull distribution with
application to failure data Journal of the Franklin
Institute Vol 347 No 8 1399-1429
httpdxdoiorg101016jjfranklin201006010
[10] de Pascoa M A R Ortega E M M amp Cordeiro G
M (2011) The Kumaraswamy generalized gamma
distribution with application in survival analysis
Statistical Methodology 8 411ndash433
wwwdoi101016jstamet201104001
[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon
Probability Distribution and its Application in
Reliability Theory Acta PolytechnicaHungarica Vol
15 No 1 pp 197 ndash 216
[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal
distribution and its applications Communication in
Statistics Theory and Methods Vol 31 497-512
[13] Gomes A E da-Silva C Q Cordeiro G M amp
Ortega E M (2014) A new lifetime model the
Kumaraswamy generalized Rayleigh distribution
Journal of statistical computation and simulation 84
290-309
[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001
[15] Klakattawi H S (2019) The Weibull-Gamma
Distribution Properties and Applications Entropy 21
438 pp 1 ndash 15 doi103390e21050438
[16] Kumaraswamy P (1980) A generalized probability
density function for double-bounded random processes
Journal of Hydrology Vol 46 79-88
httpdxdoiorg1010160022-1694(80)90036-0
[17] Nadarajah S (2008) On the distribution of
Kumaraswamy Journal of Hydrology 348 568-569
`httpdxdoiorg101016jjhydrol200709008
[18] Oguntunde P E Babatunde O S ampOgunmola A O
(2014) Theoretical Analysis of the Kumaraswamy-
Inverse Exponential Distribution International Journal
of Statistics and Applications Vol 4 No 2 113-116
doi105923jstatistics2014040205
[19] Zografos K amp Balakrishnan N (2009) On families of
beta and generalized gamma generated distributions and
associated inference Statistical Methodology 6 344 ndash
362
Monthly tax revenue0 10 20 30 40
Densi
ty0
00
00
40
08
00
60
02
Paper ID ART20202623 1021275ART20202623 1203
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Appendix
Proof of Proposition 1
Proof
(i) We need to find the limiting values of 119891119883 119909 That is
lim119909rarr0
119891119883 119909 = lim119909rarr0
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr0
1205752
1205752 minus 1199092 lim119909rarr0
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr0
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr0
1minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ 1 ∙ 1 ∙ 1 minus 1 119886minus1 ∙ 1 minus 1 minus 1 119886 119887minus1
= 0
Also
lim119909rarr120575
119891119883 119909 = lim119909rarr120575
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr120575
1205752
1205752 minus 1199092 lim119909rarr120575
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr120575
1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr120575
1minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ infin ∙ 0 ∙ 1 minus 0 119886minus1 ∙ 1 minus 1 minus 0 119886 119887minus1
= 0
Since 119891119883 119909 at both limits is zero (0) it is unimodal
(ii) 119891119883 119909 120575
0119889119909 = lim119909rarr120575 119891119883 119905
119909
0119889119905
= lim119909rarr120575
119865119883 119909
= lim119909rarr120575
1minus 1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus lim119909rarr120575
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus 1minus 1minus 120575 + 120575
120575 minus 120575 minus120582
2120575
119886
119887
= 1minus 1 minus 1 minus 0 119886 119887
= 1
Hourly record of deuterium excesses from 2015-11-11T0000 to 2015-11-24T2300
302 257 174 225 155 066 056 092 124 120 086 118 134 194 197 179 182 194 206 184 224 201
232 315 345 338 354 389 498 343 398 419 436 497 465 415 349 319 337 358 382 400 421 469
486 506 493 477 494 495 534 526 476 535 559 554 578 592 580 569 585 574 564 582 521 548
619 792 893 845 874 835 840 800 769 622 699 685 745 788 780 805 822 840 790 665 629 674
572 475 412 420 386 352 353 327 323 313 262 394 452 512 577 594 592 586 586 579 535 557
571 581 596 604 658 669 670 697 699 723 732 661 720 799 817 850 858 851 897 888 873 862
893 945 912 909 873 898 876 891 918 930 923 907 859 908 947 912 873 847 889 815 677 672
641 617 479 470 435 432 432 411 398 425 421 397 351 270 256 267 212 258 237 199 157 120
119 070 094 107 109 131 118 086 081 047 070 083 104 169 065 133 155 213 238 267 249 219
220 169 166 187 267 243 193 263 307 361 340 314 354 293 339 245 250 341 329 357 341 367
303 337 371 374 384 352 347 375 365 348 349 440 367 441 365 349 285 379 386 400 388 469
427 380 372 389 371 293 210 179 298 391 528 551 583 554 542 493 450 312 433 460 400 430
485 501 490 550 607 636 650 683 739 756 722 728 787 763 768 700 728 719 686 691 664672
687 714 686 680 630 623 590 602 616 585 598 555 553 545 561 531 524 492 527 224 250 257
Monthly records of tax revenue of Egypt between January 2006 and November 2010 (1000 million Egyptian pounds)
59 204 149 162 172 78 61 92 102 96 133 85 216 185 51 67 17 86 97 392 357 157 97 1041 36
85 8 92 262 219 167 213 354 143 85 106 191 205 71 77 181 165 119 7 86 125103 112 61 84
11 116 119 52 68 89 7110
Paper ID ART20202623 1021275ART20202623 1204
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
whereΥ = 1205752
1205752minus1199092 Ζ = 1 minus 119909+120575
120575minus119909 minus120582
2120575
0 lt 119909 lt 120575119886 119887 120575 120582 gt
0
The skewness and tail weight of the distribution are
controlled by the parameters 119886 and 119887 The plots of the K-
epsilon density function (8) at various parameter values are
given in Fig 1 below
Figure 1 K-Epsilon probability density plots at various parameter values
From (7) the quantile function of the K-epsilon distribution
is given by
where 0 lt 119901 lt 1 119886 119887 120575 120582 gt 0
Proposition 1 The K-epsilon distribution is a true probability density
function with unimodal shape property That is we want to
show that
(i) 119891119883 119909 is unimodal and
(ii) 119891119883 119909 120575
0119889119909 = 1
The proof of this proposition is given in the appendix
31 Moments of the K-epsilon distribution
The 119903119905119893 119903 = 1 2hellip moment of the K-epsilon can be
derived from the expression
From (10) it is obvious that the moments of the K-epsilon
distribution cannot be obtained in explicit form but can be
obtained by numerical integration for the desired moment(s)
when the parameter values are specified
32 Survival and hazard rate functions
Survival and hazard rate functions form the focal points in
survival analysis and reliability theory The survival
function which is the complementary cumulative
distribution or reliability function is defined as the
probability of surviving to the age of 119909 or exceeding age 119909
On the other hand the hazard rate function also known as
instantaneous failure rate or force of mortality is interpreted
as the amount of risk of failure associated with a unit of age
119909 The two functions are related however the hazard rate
a = 2 b = 04 120575 120582 = 4 (black) 25 (red)
15 (blue) 05 (green)
0 1 2 3 4 5 6X values
00
02
04
06
08
Den
sity
a = 2 120582 = 3 120575 = 6
b = 015 (black) 025 (red)
035 (blue) 05 (green)
0 1 2 3 4 5 6X values
00
02
04
06
08
Den
sity
a = 2 b = 04 120582 120575 = 6 (black) 10 (red)
15 (blue) 25 (green)
0 1 2 3 4 5 6X
values
00
02
04
06
08
Den
sity
0 1 2 3 4 5 6X values
00
02
04
06
08
Den
sity
b = 04 120582 = 3 120575 = 6
a = 2 (black) 3 (red)
5 (blue) 10 (green)
Paper ID ART20202623 1021275ART20202623 1200
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
function is more informative about the underlying
mechanism of failure
The survival function of the K-epsilon distribution is given
by
and its hazard rate function is given by
The plots of the hazard rate function of the K-epsilon
distribution at various parameter values are presented in Fig
2 below The shapes suggest that the K-epsilon hazard rate
function could be a good choice model for somefailure rate
datasets
Figure 2 K-Epsilon distribution hazard rate function at various parameter values
33 Order statistics distribution
Order statistics play important role in life testing and
reliability analysis Given a random sample 1199091 1199092 hellip 119909119899 of
a random variable 119883 having the K-epsilon distribution and
density functions 119865119883 and 119891119883 respectively The distribution of
the 119903119905119893order statistic 119883(119903) of the ordered sample 119909(1) hellip
119909(119903) hellip 119909(119899) is given by
119891119883 119903 119909 = 119886119887120582ΛΥ 1 minus Ζ Ζ119886minus1Ψ119887 119899minus119903+1 minus1 1 minusΨ119887 119903minus1(13)
Where Λ =119899
119903minus1 119899minus119903 Υ =
1205752
1205752minus1199092
Ζ = 1 minus 119909+120575
120575minus119909 minus120582
2120575
andΨ = 1minus Ζ119886 From (13) the K-epsilon
distribution of the first order statistic 119883(1) and the 119899119905119893 order
statistic 119883(119899) are given respectively by
4 Parameter Estimation
As noted earlier the explicit expressions for the K-epsilon
probability density cumulative distribution and quantile
functions has made the distribution a good candidate for the
deployment of the fitdistrplus package in R This has the
advantage of producing parameter estimates their standard
errors and many other important statistical properties
5 Application
The K-epsilon distribution is applied to 2 datasets
namelydeuterium excesses and tax revenue datasets These
are represented in boxplots in Fig 3 below
Paper ID ART20202623 1021275ART20202623 1201
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 3 Boxplots of Deuterium excesses and Monthly Tax
Revenue
The first dataset is hourly record of deuterium (heavy
hydrogen ndash one of the two stable isotopes of hydrogen)
excesses from 2015-11-11T0000 to 2015-11-24T2300 [2]
given in the appendix
Parameter estimates and Kolmogorov-Smirnov goodness-of-
fit test results are given in Table 1 while the fitted K-epsilon
densityplot is presented in Fig 3
Table 1 K-epsilon distribution fit results for deuterium excesses dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark
K-epsilon 15866 (01061) 22124 (959727) 00050 (00007) 97456 (02744) 67668 00518 (00781) Good fit
119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value
From Table 1 it could be observed that the K-epsilon
distribution has fitted the deuterium excesses datasetThe
results of the goodness-of-fit test shown in Table 1 above
provide sufficient evidence that the K-epsilon distribution
fits the dataset Also the standard errors of the parameter
estimates obtained indicate they are of good precision
Figure 3 K-Epsilon distribution fit of hydrogen
Isotope (Deuterium) excesses
The second dataset for application is the monthly records of
tax revenue of Egypt between January 2006 and November
2010 in 1000 million Egyptian pounds reported in [15] It is
presented in the appendix
The results for parameter estimate and goodness-of-fit test
are presented in Table 2 while the fitted K-epsilon
probability density plot is presented in Fig 4
Table 2 K-epsilon distribution fit results for Monthly Tax Revenue dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark
K-epsilon 638712 (18779) 01877 (00378) 06924 (00637) 20488 (02086) 187973 00735 (01771) Good fit
119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value
Table 2 also suggest that the K-epsilon distribution has
successfully fitted the tax revenue dataset The parameter
estimates are of good precision
Den
sity
00
00
05
01
00
15
Deuterium excesses0 2 4 6 8 10
Paper ID ART20202623 1021275ART20202623 1202
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 5 K-epsilon distribution fit of Monthly Tax Revenue
6 Conclusion
A new distribution called the Kumaraswamy epsilon (K-
epsilon) distribution has been derived in this study It is
constructed from the Kumaraswamy generator
(Kumaraswamy-G) of probability density functions Here
the parent distribution is the epsilon distribution The new
distribution unlike the beta distribution has explicit
expressions for the cumulative distribution and quantile
functions This makes parameter estimates much easier
particularly in R Although the moments of the K-epsilon
distribution could not be expressed in explicit form other
properties of the K-epsilon distribution such as the
distribution of order statistics survival and hazard rate
functions have been derived The shapes of the plots of the
probability density and hazard rate functions at various
parameter values suggest that the K-epsilon distribution
holds the potential for wide applications within its parameter
space
The K-epsilon distribution was applied to two different
lifetime datasets namely deuterium (hydrogen isotope)
excesses and monthly tax revenue The results of the fit of
this distribution to these datasets are good confirming its
wide usefulness in practical applications
References
[1] Aryal G (2015) Kumaraswamy Laplace Distribution
and its Applications Paper presented at the Seventh
International Conference on Dynamic Systems and
Applications and Fifth International Conference on
Neural Parallel and Scientific Computations
[2] Bonne J-L Werner M Meyer H Kipfstuhl S
Rabe B Behrens Melanie K SchAtildeparanicke L amp
Steen-Larsen H-C (2019) Water vapour isotopes
analyser calibrated data from POLARSTERN during
dockyard 2015-08 at Port of TromsAtildecedil Norway
PANGAEA httpsdoiorg101594PANGAEA897410
[3] Bourguignon M Silva R B amp Cordeiro G M (2014)
The Weibull-G family of probability distributions
Journal of Data Science 12 53 ndash 68
[4] Bourguignon M Silva R B Zea L M amp Cordeiro
G M (2013) The Kumaraswamy Pareto distribution
Journal of Statistical Theory and Applications 12 129-
144
[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new
customer lifetime duration distribution the
Kumaraswamy Lindley distribution International
Journal of Trade Economics and Finance Vol 5 No
5 441 ndash 444
[6] Cordeiro G M amp de Castro M (2011) A new family
of generalized distributions Journal of statistical
computation and simulation 81 883-898
httpdxdoiorg10108000949650903530745
[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)
The Kumaraswamy Gumbel distribution Statistical
Methods and Applications 21 139-168
[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27
[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)
The Kumaraswamy Weibull distribution with
application to failure data Journal of the Franklin
Institute Vol 347 No 8 1399-1429
httpdxdoiorg101016jjfranklin201006010
[10] de Pascoa M A R Ortega E M M amp Cordeiro G
M (2011) The Kumaraswamy generalized gamma
distribution with application in survival analysis
Statistical Methodology 8 411ndash433
wwwdoi101016jstamet201104001
[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon
Probability Distribution and its Application in
Reliability Theory Acta PolytechnicaHungarica Vol
15 No 1 pp 197 ndash 216
[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal
distribution and its applications Communication in
Statistics Theory and Methods Vol 31 497-512
[13] Gomes A E da-Silva C Q Cordeiro G M amp
Ortega E M (2014) A new lifetime model the
Kumaraswamy generalized Rayleigh distribution
Journal of statistical computation and simulation 84
290-309
[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001
[15] Klakattawi H S (2019) The Weibull-Gamma
Distribution Properties and Applications Entropy 21
438 pp 1 ndash 15 doi103390e21050438
[16] Kumaraswamy P (1980) A generalized probability
density function for double-bounded random processes
Journal of Hydrology Vol 46 79-88
httpdxdoiorg1010160022-1694(80)90036-0
[17] Nadarajah S (2008) On the distribution of
Kumaraswamy Journal of Hydrology 348 568-569
`httpdxdoiorg101016jjhydrol200709008
[18] Oguntunde P E Babatunde O S ampOgunmola A O
(2014) Theoretical Analysis of the Kumaraswamy-
Inverse Exponential Distribution International Journal
of Statistics and Applications Vol 4 No 2 113-116
doi105923jstatistics2014040205
[19] Zografos K amp Balakrishnan N (2009) On families of
beta and generalized gamma generated distributions and
associated inference Statistical Methodology 6 344 ndash
362
Monthly tax revenue0 10 20 30 40
Densi
ty0
00
00
40
08
00
60
02
Paper ID ART20202623 1021275ART20202623 1203
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Appendix
Proof of Proposition 1
Proof
(i) We need to find the limiting values of 119891119883 119909 That is
lim119909rarr0
119891119883 119909 = lim119909rarr0
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr0
1205752
1205752 minus 1199092 lim119909rarr0
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr0
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr0
1minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ 1 ∙ 1 ∙ 1 minus 1 119886minus1 ∙ 1 minus 1 minus 1 119886 119887minus1
= 0
Also
lim119909rarr120575
119891119883 119909 = lim119909rarr120575
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr120575
1205752
1205752 minus 1199092 lim119909rarr120575
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr120575
1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr120575
1minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ infin ∙ 0 ∙ 1 minus 0 119886minus1 ∙ 1 minus 1 minus 0 119886 119887minus1
= 0
Since 119891119883 119909 at both limits is zero (0) it is unimodal
(ii) 119891119883 119909 120575
0119889119909 = lim119909rarr120575 119891119883 119905
119909
0119889119905
= lim119909rarr120575
119865119883 119909
= lim119909rarr120575
1minus 1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus lim119909rarr120575
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus 1minus 1minus 120575 + 120575
120575 minus 120575 minus120582
2120575
119886
119887
= 1minus 1 minus 1 minus 0 119886 119887
= 1
Hourly record of deuterium excesses from 2015-11-11T0000 to 2015-11-24T2300
302 257 174 225 155 066 056 092 124 120 086 118 134 194 197 179 182 194 206 184 224 201
232 315 345 338 354 389 498 343 398 419 436 497 465 415 349 319 337 358 382 400 421 469
486 506 493 477 494 495 534 526 476 535 559 554 578 592 580 569 585 574 564 582 521 548
619 792 893 845 874 835 840 800 769 622 699 685 745 788 780 805 822 840 790 665 629 674
572 475 412 420 386 352 353 327 323 313 262 394 452 512 577 594 592 586 586 579 535 557
571 581 596 604 658 669 670 697 699 723 732 661 720 799 817 850 858 851 897 888 873 862
893 945 912 909 873 898 876 891 918 930 923 907 859 908 947 912 873 847 889 815 677 672
641 617 479 470 435 432 432 411 398 425 421 397 351 270 256 267 212 258 237 199 157 120
119 070 094 107 109 131 118 086 081 047 070 083 104 169 065 133 155 213 238 267 249 219
220 169 166 187 267 243 193 263 307 361 340 314 354 293 339 245 250 341 329 357 341 367
303 337 371 374 384 352 347 375 365 348 349 440 367 441 365 349 285 379 386 400 388 469
427 380 372 389 371 293 210 179 298 391 528 551 583 554 542 493 450 312 433 460 400 430
485 501 490 550 607 636 650 683 739 756 722 728 787 763 768 700 728 719 686 691 664672
687 714 686 680 630 623 590 602 616 585 598 555 553 545 561 531 524 492 527 224 250 257
Monthly records of tax revenue of Egypt between January 2006 and November 2010 (1000 million Egyptian pounds)
59 204 149 162 172 78 61 92 102 96 133 85 216 185 51 67 17 86 97 392 357 157 97 1041 36
85 8 92 262 219 167 213 354 143 85 106 191 205 71 77 181 165 119 7 86 125103 112 61 84
11 116 119 52 68 89 7110
Paper ID ART20202623 1021275ART20202623 1204
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
function is more informative about the underlying
mechanism of failure
The survival function of the K-epsilon distribution is given
by
and its hazard rate function is given by
The plots of the hazard rate function of the K-epsilon
distribution at various parameter values are presented in Fig
2 below The shapes suggest that the K-epsilon hazard rate
function could be a good choice model for somefailure rate
datasets
Figure 2 K-Epsilon distribution hazard rate function at various parameter values
33 Order statistics distribution
Order statistics play important role in life testing and
reliability analysis Given a random sample 1199091 1199092 hellip 119909119899 of
a random variable 119883 having the K-epsilon distribution and
density functions 119865119883 and 119891119883 respectively The distribution of
the 119903119905119893order statistic 119883(119903) of the ordered sample 119909(1) hellip
119909(119903) hellip 119909(119899) is given by
119891119883 119903 119909 = 119886119887120582ΛΥ 1 minus Ζ Ζ119886minus1Ψ119887 119899minus119903+1 minus1 1 minusΨ119887 119903minus1(13)
Where Λ =119899
119903minus1 119899minus119903 Υ =
1205752
1205752minus1199092
Ζ = 1 minus 119909+120575
120575minus119909 minus120582
2120575
andΨ = 1minus Ζ119886 From (13) the K-epsilon
distribution of the first order statistic 119883(1) and the 119899119905119893 order
statistic 119883(119899) are given respectively by
4 Parameter Estimation
As noted earlier the explicit expressions for the K-epsilon
probability density cumulative distribution and quantile
functions has made the distribution a good candidate for the
deployment of the fitdistrplus package in R This has the
advantage of producing parameter estimates their standard
errors and many other important statistical properties
5 Application
The K-epsilon distribution is applied to 2 datasets
namelydeuterium excesses and tax revenue datasets These
are represented in boxplots in Fig 3 below
Paper ID ART20202623 1021275ART20202623 1201
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 3 Boxplots of Deuterium excesses and Monthly Tax
Revenue
The first dataset is hourly record of deuterium (heavy
hydrogen ndash one of the two stable isotopes of hydrogen)
excesses from 2015-11-11T0000 to 2015-11-24T2300 [2]
given in the appendix
Parameter estimates and Kolmogorov-Smirnov goodness-of-
fit test results are given in Table 1 while the fitted K-epsilon
densityplot is presented in Fig 3
Table 1 K-epsilon distribution fit results for deuterium excesses dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark
K-epsilon 15866 (01061) 22124 (959727) 00050 (00007) 97456 (02744) 67668 00518 (00781) Good fit
119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value
From Table 1 it could be observed that the K-epsilon
distribution has fitted the deuterium excesses datasetThe
results of the goodness-of-fit test shown in Table 1 above
provide sufficient evidence that the K-epsilon distribution
fits the dataset Also the standard errors of the parameter
estimates obtained indicate they are of good precision
Figure 3 K-Epsilon distribution fit of hydrogen
Isotope (Deuterium) excesses
The second dataset for application is the monthly records of
tax revenue of Egypt between January 2006 and November
2010 in 1000 million Egyptian pounds reported in [15] It is
presented in the appendix
The results for parameter estimate and goodness-of-fit test
are presented in Table 2 while the fitted K-epsilon
probability density plot is presented in Fig 4
Table 2 K-epsilon distribution fit results for Monthly Tax Revenue dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark
K-epsilon 638712 (18779) 01877 (00378) 06924 (00637) 20488 (02086) 187973 00735 (01771) Good fit
119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value
Table 2 also suggest that the K-epsilon distribution has
successfully fitted the tax revenue dataset The parameter
estimates are of good precision
Den
sity
00
00
05
01
00
15
Deuterium excesses0 2 4 6 8 10
Paper ID ART20202623 1021275ART20202623 1202
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 5 K-epsilon distribution fit of Monthly Tax Revenue
6 Conclusion
A new distribution called the Kumaraswamy epsilon (K-
epsilon) distribution has been derived in this study It is
constructed from the Kumaraswamy generator
(Kumaraswamy-G) of probability density functions Here
the parent distribution is the epsilon distribution The new
distribution unlike the beta distribution has explicit
expressions for the cumulative distribution and quantile
functions This makes parameter estimates much easier
particularly in R Although the moments of the K-epsilon
distribution could not be expressed in explicit form other
properties of the K-epsilon distribution such as the
distribution of order statistics survival and hazard rate
functions have been derived The shapes of the plots of the
probability density and hazard rate functions at various
parameter values suggest that the K-epsilon distribution
holds the potential for wide applications within its parameter
space
The K-epsilon distribution was applied to two different
lifetime datasets namely deuterium (hydrogen isotope)
excesses and monthly tax revenue The results of the fit of
this distribution to these datasets are good confirming its
wide usefulness in practical applications
References
[1] Aryal G (2015) Kumaraswamy Laplace Distribution
and its Applications Paper presented at the Seventh
International Conference on Dynamic Systems and
Applications and Fifth International Conference on
Neural Parallel and Scientific Computations
[2] Bonne J-L Werner M Meyer H Kipfstuhl S
Rabe B Behrens Melanie K SchAtildeparanicke L amp
Steen-Larsen H-C (2019) Water vapour isotopes
analyser calibrated data from POLARSTERN during
dockyard 2015-08 at Port of TromsAtildecedil Norway
PANGAEA httpsdoiorg101594PANGAEA897410
[3] Bourguignon M Silva R B amp Cordeiro G M (2014)
The Weibull-G family of probability distributions
Journal of Data Science 12 53 ndash 68
[4] Bourguignon M Silva R B Zea L M amp Cordeiro
G M (2013) The Kumaraswamy Pareto distribution
Journal of Statistical Theory and Applications 12 129-
144
[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new
customer lifetime duration distribution the
Kumaraswamy Lindley distribution International
Journal of Trade Economics and Finance Vol 5 No
5 441 ndash 444
[6] Cordeiro G M amp de Castro M (2011) A new family
of generalized distributions Journal of statistical
computation and simulation 81 883-898
httpdxdoiorg10108000949650903530745
[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)
The Kumaraswamy Gumbel distribution Statistical
Methods and Applications 21 139-168
[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27
[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)
The Kumaraswamy Weibull distribution with
application to failure data Journal of the Franklin
Institute Vol 347 No 8 1399-1429
httpdxdoiorg101016jjfranklin201006010
[10] de Pascoa M A R Ortega E M M amp Cordeiro G
M (2011) The Kumaraswamy generalized gamma
distribution with application in survival analysis
Statistical Methodology 8 411ndash433
wwwdoi101016jstamet201104001
[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon
Probability Distribution and its Application in
Reliability Theory Acta PolytechnicaHungarica Vol
15 No 1 pp 197 ndash 216
[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal
distribution and its applications Communication in
Statistics Theory and Methods Vol 31 497-512
[13] Gomes A E da-Silva C Q Cordeiro G M amp
Ortega E M (2014) A new lifetime model the
Kumaraswamy generalized Rayleigh distribution
Journal of statistical computation and simulation 84
290-309
[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001
[15] Klakattawi H S (2019) The Weibull-Gamma
Distribution Properties and Applications Entropy 21
438 pp 1 ndash 15 doi103390e21050438
[16] Kumaraswamy P (1980) A generalized probability
density function for double-bounded random processes
Journal of Hydrology Vol 46 79-88
httpdxdoiorg1010160022-1694(80)90036-0
[17] Nadarajah S (2008) On the distribution of
Kumaraswamy Journal of Hydrology 348 568-569
`httpdxdoiorg101016jjhydrol200709008
[18] Oguntunde P E Babatunde O S ampOgunmola A O
(2014) Theoretical Analysis of the Kumaraswamy-
Inverse Exponential Distribution International Journal
of Statistics and Applications Vol 4 No 2 113-116
doi105923jstatistics2014040205
[19] Zografos K amp Balakrishnan N (2009) On families of
beta and generalized gamma generated distributions and
associated inference Statistical Methodology 6 344 ndash
362
Monthly tax revenue0 10 20 30 40
Densi
ty0
00
00
40
08
00
60
02
Paper ID ART20202623 1021275ART20202623 1203
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Appendix
Proof of Proposition 1
Proof
(i) We need to find the limiting values of 119891119883 119909 That is
lim119909rarr0
119891119883 119909 = lim119909rarr0
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr0
1205752
1205752 minus 1199092 lim119909rarr0
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr0
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr0
1minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ 1 ∙ 1 ∙ 1 minus 1 119886minus1 ∙ 1 minus 1 minus 1 119886 119887minus1
= 0
Also
lim119909rarr120575
119891119883 119909 = lim119909rarr120575
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr120575
1205752
1205752 minus 1199092 lim119909rarr120575
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr120575
1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr120575
1minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ infin ∙ 0 ∙ 1 minus 0 119886minus1 ∙ 1 minus 1 minus 0 119886 119887minus1
= 0
Since 119891119883 119909 at both limits is zero (0) it is unimodal
(ii) 119891119883 119909 120575
0119889119909 = lim119909rarr120575 119891119883 119905
119909
0119889119905
= lim119909rarr120575
119865119883 119909
= lim119909rarr120575
1minus 1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus lim119909rarr120575
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus 1minus 1minus 120575 + 120575
120575 minus 120575 minus120582
2120575
119886
119887
= 1minus 1 minus 1 minus 0 119886 119887
= 1
Hourly record of deuterium excesses from 2015-11-11T0000 to 2015-11-24T2300
302 257 174 225 155 066 056 092 124 120 086 118 134 194 197 179 182 194 206 184 224 201
232 315 345 338 354 389 498 343 398 419 436 497 465 415 349 319 337 358 382 400 421 469
486 506 493 477 494 495 534 526 476 535 559 554 578 592 580 569 585 574 564 582 521 548
619 792 893 845 874 835 840 800 769 622 699 685 745 788 780 805 822 840 790 665 629 674
572 475 412 420 386 352 353 327 323 313 262 394 452 512 577 594 592 586 586 579 535 557
571 581 596 604 658 669 670 697 699 723 732 661 720 799 817 850 858 851 897 888 873 862
893 945 912 909 873 898 876 891 918 930 923 907 859 908 947 912 873 847 889 815 677 672
641 617 479 470 435 432 432 411 398 425 421 397 351 270 256 267 212 258 237 199 157 120
119 070 094 107 109 131 118 086 081 047 070 083 104 169 065 133 155 213 238 267 249 219
220 169 166 187 267 243 193 263 307 361 340 314 354 293 339 245 250 341 329 357 341 367
303 337 371 374 384 352 347 375 365 348 349 440 367 441 365 349 285 379 386 400 388 469
427 380 372 389 371 293 210 179 298 391 528 551 583 554 542 493 450 312 433 460 400 430
485 501 490 550 607 636 650 683 739 756 722 728 787 763 768 700 728 719 686 691 664672
687 714 686 680 630 623 590 602 616 585 598 555 553 545 561 531 524 492 527 224 250 257
Monthly records of tax revenue of Egypt between January 2006 and November 2010 (1000 million Egyptian pounds)
59 204 149 162 172 78 61 92 102 96 133 85 216 185 51 67 17 86 97 392 357 157 97 1041 36
85 8 92 262 219 167 213 354 143 85 106 191 205 71 77 181 165 119 7 86 125103 112 61 84
11 116 119 52 68 89 7110
Paper ID ART20202623 1021275ART20202623 1204
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 3 Boxplots of Deuterium excesses and Monthly Tax
Revenue
The first dataset is hourly record of deuterium (heavy
hydrogen ndash one of the two stable isotopes of hydrogen)
excesses from 2015-11-11T0000 to 2015-11-24T2300 [2]
given in the appendix
Parameter estimates and Kolmogorov-Smirnov goodness-of-
fit test results are given in Table 1 while the fitted K-epsilon
densityplot is presented in Fig 3
Table 1 K-epsilon distribution fit results for deuterium excesses dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark
K-epsilon 15866 (01061) 22124 (959727) 00050 (00007) 97456 (02744) 67668 00518 (00781) Good fit
119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value
From Table 1 it could be observed that the K-epsilon
distribution has fitted the deuterium excesses datasetThe
results of the goodness-of-fit test shown in Table 1 above
provide sufficient evidence that the K-epsilon distribution
fits the dataset Also the standard errors of the parameter
estimates obtained indicate they are of good precision
Figure 3 K-Epsilon distribution fit of hydrogen
Isotope (Deuterium) excesses
The second dataset for application is the monthly records of
tax revenue of Egypt between January 2006 and November
2010 in 1000 million Egyptian pounds reported in [15] It is
presented in the appendix
The results for parameter estimate and goodness-of-fit test
are presented in Table 2 while the fitted K-epsilon
probability density plot is presented in Fig 4
Table 2 K-epsilon distribution fit results for Monthly Tax Revenue dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark
K-epsilon 638712 (18779) 01877 (00378) 06924 (00637) 20488 (02086) 187973 00735 (01771) Good fit
119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value
Table 2 also suggest that the K-epsilon distribution has
successfully fitted the tax revenue dataset The parameter
estimates are of good precision
Den
sity
00
00
05
01
00
15
Deuterium excesses0 2 4 6 8 10
Paper ID ART20202623 1021275ART20202623 1202
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 5 K-epsilon distribution fit of Monthly Tax Revenue
6 Conclusion
A new distribution called the Kumaraswamy epsilon (K-
epsilon) distribution has been derived in this study It is
constructed from the Kumaraswamy generator
(Kumaraswamy-G) of probability density functions Here
the parent distribution is the epsilon distribution The new
distribution unlike the beta distribution has explicit
expressions for the cumulative distribution and quantile
functions This makes parameter estimates much easier
particularly in R Although the moments of the K-epsilon
distribution could not be expressed in explicit form other
properties of the K-epsilon distribution such as the
distribution of order statistics survival and hazard rate
functions have been derived The shapes of the plots of the
probability density and hazard rate functions at various
parameter values suggest that the K-epsilon distribution
holds the potential for wide applications within its parameter
space
The K-epsilon distribution was applied to two different
lifetime datasets namely deuterium (hydrogen isotope)
excesses and monthly tax revenue The results of the fit of
this distribution to these datasets are good confirming its
wide usefulness in practical applications
References
[1] Aryal G (2015) Kumaraswamy Laplace Distribution
and its Applications Paper presented at the Seventh
International Conference on Dynamic Systems and
Applications and Fifth International Conference on
Neural Parallel and Scientific Computations
[2] Bonne J-L Werner M Meyer H Kipfstuhl S
Rabe B Behrens Melanie K SchAtildeparanicke L amp
Steen-Larsen H-C (2019) Water vapour isotopes
analyser calibrated data from POLARSTERN during
dockyard 2015-08 at Port of TromsAtildecedil Norway
PANGAEA httpsdoiorg101594PANGAEA897410
[3] Bourguignon M Silva R B amp Cordeiro G M (2014)
The Weibull-G family of probability distributions
Journal of Data Science 12 53 ndash 68
[4] Bourguignon M Silva R B Zea L M amp Cordeiro
G M (2013) The Kumaraswamy Pareto distribution
Journal of Statistical Theory and Applications 12 129-
144
[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new
customer lifetime duration distribution the
Kumaraswamy Lindley distribution International
Journal of Trade Economics and Finance Vol 5 No
5 441 ndash 444
[6] Cordeiro G M amp de Castro M (2011) A new family
of generalized distributions Journal of statistical
computation and simulation 81 883-898
httpdxdoiorg10108000949650903530745
[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)
The Kumaraswamy Gumbel distribution Statistical
Methods and Applications 21 139-168
[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27
[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)
The Kumaraswamy Weibull distribution with
application to failure data Journal of the Franklin
Institute Vol 347 No 8 1399-1429
httpdxdoiorg101016jjfranklin201006010
[10] de Pascoa M A R Ortega E M M amp Cordeiro G
M (2011) The Kumaraswamy generalized gamma
distribution with application in survival analysis
Statistical Methodology 8 411ndash433
wwwdoi101016jstamet201104001
[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon
Probability Distribution and its Application in
Reliability Theory Acta PolytechnicaHungarica Vol
15 No 1 pp 197 ndash 216
[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal
distribution and its applications Communication in
Statistics Theory and Methods Vol 31 497-512
[13] Gomes A E da-Silva C Q Cordeiro G M amp
Ortega E M (2014) A new lifetime model the
Kumaraswamy generalized Rayleigh distribution
Journal of statistical computation and simulation 84
290-309
[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001
[15] Klakattawi H S (2019) The Weibull-Gamma
Distribution Properties and Applications Entropy 21
438 pp 1 ndash 15 doi103390e21050438
[16] Kumaraswamy P (1980) A generalized probability
density function for double-bounded random processes
Journal of Hydrology Vol 46 79-88
httpdxdoiorg1010160022-1694(80)90036-0
[17] Nadarajah S (2008) On the distribution of
Kumaraswamy Journal of Hydrology 348 568-569
`httpdxdoiorg101016jjhydrol200709008
[18] Oguntunde P E Babatunde O S ampOgunmola A O
(2014) Theoretical Analysis of the Kumaraswamy-
Inverse Exponential Distribution International Journal
of Statistics and Applications Vol 4 No 2 113-116
doi105923jstatistics2014040205
[19] Zografos K amp Balakrishnan N (2009) On families of
beta and generalized gamma generated distributions and
associated inference Statistical Methodology 6 344 ndash
362
Monthly tax revenue0 10 20 30 40
Densi
ty0
00
00
40
08
00
60
02
Paper ID ART20202623 1021275ART20202623 1203
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Appendix
Proof of Proposition 1
Proof
(i) We need to find the limiting values of 119891119883 119909 That is
lim119909rarr0
119891119883 119909 = lim119909rarr0
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr0
1205752
1205752 minus 1199092 lim119909rarr0
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr0
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr0
1minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ 1 ∙ 1 ∙ 1 minus 1 119886minus1 ∙ 1 minus 1 minus 1 119886 119887minus1
= 0
Also
lim119909rarr120575
119891119883 119909 = lim119909rarr120575
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr120575
1205752
1205752 minus 1199092 lim119909rarr120575
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr120575
1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr120575
1minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ infin ∙ 0 ∙ 1 minus 0 119886minus1 ∙ 1 minus 1 minus 0 119886 119887minus1
= 0
Since 119891119883 119909 at both limits is zero (0) it is unimodal
(ii) 119891119883 119909 120575
0119889119909 = lim119909rarr120575 119891119883 119905
119909
0119889119905
= lim119909rarr120575
119865119883 119909
= lim119909rarr120575
1minus 1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus lim119909rarr120575
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus 1minus 1minus 120575 + 120575
120575 minus 120575 minus120582
2120575
119886
119887
= 1minus 1 minus 1 minus 0 119886 119887
= 1
Hourly record of deuterium excesses from 2015-11-11T0000 to 2015-11-24T2300
302 257 174 225 155 066 056 092 124 120 086 118 134 194 197 179 182 194 206 184 224 201
232 315 345 338 354 389 498 343 398 419 436 497 465 415 349 319 337 358 382 400 421 469
486 506 493 477 494 495 534 526 476 535 559 554 578 592 580 569 585 574 564 582 521 548
619 792 893 845 874 835 840 800 769 622 699 685 745 788 780 805 822 840 790 665 629 674
572 475 412 420 386 352 353 327 323 313 262 394 452 512 577 594 592 586 586 579 535 557
571 581 596 604 658 669 670 697 699 723 732 661 720 799 817 850 858 851 897 888 873 862
893 945 912 909 873 898 876 891 918 930 923 907 859 908 947 912 873 847 889 815 677 672
641 617 479 470 435 432 432 411 398 425 421 397 351 270 256 267 212 258 237 199 157 120
119 070 094 107 109 131 118 086 081 047 070 083 104 169 065 133 155 213 238 267 249 219
220 169 166 187 267 243 193 263 307 361 340 314 354 293 339 245 250 341 329 357 341 367
303 337 371 374 384 352 347 375 365 348 349 440 367 441 365 349 285 379 386 400 388 469
427 380 372 389 371 293 210 179 298 391 528 551 583 554 542 493 450 312 433 460 400 430
485 501 490 550 607 636 650 683 739 756 722 728 787 763 768 700 728 719 686 691 664672
687 714 686 680 630 623 590 602 616 585 598 555 553 545 561 531 524 492 527 224 250 257
Monthly records of tax revenue of Egypt between January 2006 and November 2010 (1000 million Egyptian pounds)
59 204 149 162 172 78 61 92 102 96 133 85 216 185 51 67 17 86 97 392 357 157 97 1041 36
85 8 92 262 219 167 213 354 143 85 106 191 205 71 77 181 165 119 7 86 125103 112 61 84
11 116 119 52 68 89 7110
Paper ID ART20202623 1021275ART20202623 1204
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Figure 5 K-epsilon distribution fit of Monthly Tax Revenue
6 Conclusion
A new distribution called the Kumaraswamy epsilon (K-
epsilon) distribution has been derived in this study It is
constructed from the Kumaraswamy generator
(Kumaraswamy-G) of probability density functions Here
the parent distribution is the epsilon distribution The new
distribution unlike the beta distribution has explicit
expressions for the cumulative distribution and quantile
functions This makes parameter estimates much easier
particularly in R Although the moments of the K-epsilon
distribution could not be expressed in explicit form other
properties of the K-epsilon distribution such as the
distribution of order statistics survival and hazard rate
functions have been derived The shapes of the plots of the
probability density and hazard rate functions at various
parameter values suggest that the K-epsilon distribution
holds the potential for wide applications within its parameter
space
The K-epsilon distribution was applied to two different
lifetime datasets namely deuterium (hydrogen isotope)
excesses and monthly tax revenue The results of the fit of
this distribution to these datasets are good confirming its
wide usefulness in practical applications
References
[1] Aryal G (2015) Kumaraswamy Laplace Distribution
and its Applications Paper presented at the Seventh
International Conference on Dynamic Systems and
Applications and Fifth International Conference on
Neural Parallel and Scientific Computations
[2] Bonne J-L Werner M Meyer H Kipfstuhl S
Rabe B Behrens Melanie K SchAtildeparanicke L amp
Steen-Larsen H-C (2019) Water vapour isotopes
analyser calibrated data from POLARSTERN during
dockyard 2015-08 at Port of TromsAtildecedil Norway
PANGAEA httpsdoiorg101594PANGAEA897410
[3] Bourguignon M Silva R B amp Cordeiro G M (2014)
The Weibull-G family of probability distributions
Journal of Data Science 12 53 ndash 68
[4] Bourguignon M Silva R B Zea L M amp Cordeiro
G M (2013) The Kumaraswamy Pareto distribution
Journal of Statistical Theory and Applications 12 129-
144
[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new
customer lifetime duration distribution the
Kumaraswamy Lindley distribution International
Journal of Trade Economics and Finance Vol 5 No
5 441 ndash 444
[6] Cordeiro G M amp de Castro M (2011) A new family
of generalized distributions Journal of statistical
computation and simulation 81 883-898
httpdxdoiorg10108000949650903530745
[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)
The Kumaraswamy Gumbel distribution Statistical
Methods and Applications 21 139-168
[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27
[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)
The Kumaraswamy Weibull distribution with
application to failure data Journal of the Franklin
Institute Vol 347 No 8 1399-1429
httpdxdoiorg101016jjfranklin201006010
[10] de Pascoa M A R Ortega E M M amp Cordeiro G
M (2011) The Kumaraswamy generalized gamma
distribution with application in survival analysis
Statistical Methodology 8 411ndash433
wwwdoi101016jstamet201104001
[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon
Probability Distribution and its Application in
Reliability Theory Acta PolytechnicaHungarica Vol
15 No 1 pp 197 ndash 216
[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal
distribution and its applications Communication in
Statistics Theory and Methods Vol 31 497-512
[13] Gomes A E da-Silva C Q Cordeiro G M amp
Ortega E M (2014) A new lifetime model the
Kumaraswamy generalized Rayleigh distribution
Journal of statistical computation and simulation 84
290-309
[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001
[15] Klakattawi H S (2019) The Weibull-Gamma
Distribution Properties and Applications Entropy 21
438 pp 1 ndash 15 doi103390e21050438
[16] Kumaraswamy P (1980) A generalized probability
density function for double-bounded random processes
Journal of Hydrology Vol 46 79-88
httpdxdoiorg1010160022-1694(80)90036-0
[17] Nadarajah S (2008) On the distribution of
Kumaraswamy Journal of Hydrology 348 568-569
`httpdxdoiorg101016jjhydrol200709008
[18] Oguntunde P E Babatunde O S ampOgunmola A O
(2014) Theoretical Analysis of the Kumaraswamy-
Inverse Exponential Distribution International Journal
of Statistics and Applications Vol 4 No 2 113-116
doi105923jstatistics2014040205
[19] Zografos K amp Balakrishnan N (2009) On families of
beta and generalized gamma generated distributions and
associated inference Statistical Methodology 6 344 ndash
362
Monthly tax revenue0 10 20 30 40
Densi
ty0
00
00
40
08
00
60
02
Paper ID ART20202623 1021275ART20202623 1203
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Appendix
Proof of Proposition 1
Proof
(i) We need to find the limiting values of 119891119883 119909 That is
lim119909rarr0
119891119883 119909 = lim119909rarr0
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr0
1205752
1205752 minus 1199092 lim119909rarr0
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr0
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr0
1minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ 1 ∙ 1 ∙ 1 minus 1 119886minus1 ∙ 1 minus 1 minus 1 119886 119887minus1
= 0
Also
lim119909rarr120575
119891119883 119909 = lim119909rarr120575
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr120575
1205752
1205752 minus 1199092 lim119909rarr120575
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr120575
1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr120575
1minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ infin ∙ 0 ∙ 1 minus 0 119886minus1 ∙ 1 minus 1 minus 0 119886 119887minus1
= 0
Since 119891119883 119909 at both limits is zero (0) it is unimodal
(ii) 119891119883 119909 120575
0119889119909 = lim119909rarr120575 119891119883 119905
119909
0119889119905
= lim119909rarr120575
119865119883 119909
= lim119909rarr120575
1minus 1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus lim119909rarr120575
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus 1minus 1minus 120575 + 120575
120575 minus 120575 minus120582
2120575
119886
119887
= 1minus 1 minus 1 minus 0 119886 119887
= 1
Hourly record of deuterium excesses from 2015-11-11T0000 to 2015-11-24T2300
302 257 174 225 155 066 056 092 124 120 086 118 134 194 197 179 182 194 206 184 224 201
232 315 345 338 354 389 498 343 398 419 436 497 465 415 349 319 337 358 382 400 421 469
486 506 493 477 494 495 534 526 476 535 559 554 578 592 580 569 585 574 564 582 521 548
619 792 893 845 874 835 840 800 769 622 699 685 745 788 780 805 822 840 790 665 629 674
572 475 412 420 386 352 353 327 323 313 262 394 452 512 577 594 592 586 586 579 535 557
571 581 596 604 658 669 670 697 699 723 732 661 720 799 817 850 858 851 897 888 873 862
893 945 912 909 873 898 876 891 918 930 923 907 859 908 947 912 873 847 889 815 677 672
641 617 479 470 435 432 432 411 398 425 421 397 351 270 256 267 212 258 237 199 157 120
119 070 094 107 109 131 118 086 081 047 070 083 104 169 065 133 155 213 238 267 249 219
220 169 166 187 267 243 193 263 307 361 340 314 354 293 339 245 250 341 329 357 341 367
303 337 371 374 384 352 347 375 365 348 349 440 367 441 365 349 285 379 386 400 388 469
427 380 372 389 371 293 210 179 298 391 528 551 583 554 542 493 450 312 433 460 400 430
485 501 490 550 607 636 650 683 739 756 722 728 787 763 768 700 728 719 686 691 664672
687 714 686 680 630 623 590 602 616 585 598 555 553 545 561 531 524 492 527 224 250 257
Monthly records of tax revenue of Egypt between January 2006 and November 2010 (1000 million Egyptian pounds)
59 204 149 162 172 78 61 92 102 96 133 85 216 185 51 67 17 86 97 392 357 157 97 1041 36
85 8 92 262 219 167 213 354 143 85 106 191 205 71 77 181 165 119 7 86 125103 112 61 84
11 116 119 52 68 89 7110
Paper ID ART20202623 1021275ART20202623 1204
International Journal of Science and Research (IJSR) ISSN 2319-7064
ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426
Volume 8 Issue 11 November 2019
wwwijsrnet Licensed Under Creative Commons Attribution CC BY
Appendix
Proof of Proposition 1
Proof
(i) We need to find the limiting values of 119891119883 119909 That is
lim119909rarr0
119891119883 119909 = lim119909rarr0
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr0
1205752
1205752 minus 1199092 lim119909rarr0
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr0
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr0
1minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ 1 ∙ 1 ∙ 1 minus 1 119886minus1 ∙ 1 minus 1 minus 1 119886 119887minus1
= 0
Also
lim119909rarr120575
119891119883 119909 = lim119909rarr120575
119886119887120582 1205752
1205752 minus 1199092 119909 + 120575
120575 minus 119909 minus120582
2120575
1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
1 minus 1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 lim119909rarr120575
1205752
1205752 minus 1199092 lim119909rarr120575
119909 + 120575
120575 minus 119909 minus120582
2120575
lim119909rarr120575
1 minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886minus1
lim119909rarr120575
1minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887minus1
= 119886119887120582 ∙ infin ∙ 0 ∙ 1 minus 0 119886minus1 ∙ 1 minus 1 minus 0 119886 119887minus1
= 0
Since 119891119883 119909 at both limits is zero (0) it is unimodal
(ii) 119891119883 119909 120575
0119889119909 = lim119909rarr120575 119891119883 119905
119909
0119889119905
= lim119909rarr120575
119865119883 119909
= lim119909rarr120575
1minus 1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus lim119909rarr120575
1 minus 1minus 119909 + 120575
120575 minus 119909 minus120582
2120575
119886
119887
= 1 minus 1minus 1minus 120575 + 120575
120575 minus 120575 minus120582
2120575
119886
119887
= 1minus 1 minus 1 minus 0 119886 119887
= 1
Hourly record of deuterium excesses from 2015-11-11T0000 to 2015-11-24T2300
302 257 174 225 155 066 056 092 124 120 086 118 134 194 197 179 182 194 206 184 224 201
232 315 345 338 354 389 498 343 398 419 436 497 465 415 349 319 337 358 382 400 421 469
486 506 493 477 494 495 534 526 476 535 559 554 578 592 580 569 585 574 564 582 521 548
619 792 893 845 874 835 840 800 769 622 699 685 745 788 780 805 822 840 790 665 629 674
572 475 412 420 386 352 353 327 323 313 262 394 452 512 577 594 592 586 586 579 535 557
571 581 596 604 658 669 670 697 699 723 732 661 720 799 817 850 858 851 897 888 873 862
893 945 912 909 873 898 876 891 918 930 923 907 859 908 947 912 873 847 889 815 677 672
641 617 479 470 435 432 432 411 398 425 421 397 351 270 256 267 212 258 237 199 157 120
119 070 094 107 109 131 118 086 081 047 070 083 104 169 065 133 155 213 238 267 249 219
220 169 166 187 267 243 193 263 307 361 340 314 354 293 339 245 250 341 329 357 341 367
303 337 371 374 384 352 347 375 365 348 349 440 367 441 365 349 285 379 386 400 388 469
427 380 372 389 371 293 210 179 298 391 528 551 583 554 542 493 450 312 433 460 400 430
485 501 490 550 607 636 650 683 739 756 722 728 787 763 768 700 728 719 686 691 664672
687 714 686 680 630 623 590 602 616 585 598 555 553 545 561 531 524 492 527 224 250 257
Monthly records of tax revenue of Egypt between January 2006 and November 2010 (1000 million Egyptian pounds)
59 204 149 162 172 78 61 92 102 96 133 85 216 185 51 67 17 86 97 392 357 157 97 1041 36
85 8 92 262 219 167 213 354 143 85 106 191 205 71 77 181 165 119 7 86 125103 112 61 84
11 116 119 52 68 89 7110
Paper ID ART20202623 1021275ART20202623 1204
Recommended