Embed Size (px)
Citation preview
1
The Beta Generalized Marshall-Olkin-G Family of Distributions
Laba Handique and Subrata Chakraborty*
Department of Statistics, Dibrugarh University
Dibrugarh-786004, India
*Corresponding Author. Email: [email protected]
(21 August 21, 2016)
Abstract
In this paper we propose a new family of distribution considering Generalized Marshal-Olkin
distribution as the base line distribution in the Beta-G family of Construction. The new family
includes Beta-G (Eugene et al. 2002 and Jones, 2004) and GMOE (Jayakumar and Mathew, 2008)
families as particular cases. Probability density function (pdf) and the cumulative distribution
function (cdf) are expressed as mixture of the Marshal-Olkin (Marshal and Olkin, 1997) distribution.
Series expansions of pdf of the order statistics are also obtained. Moments, moment generating
function, Rényi entropies, quantile power series, random sample generation and asymptotes are also
investigated. Parameter estimation by method of maximum likelihood and method of moment are
also presented. Finally proposed model is compared to the Generalized Marshall-Olkin
Kumaraswamy extended family (Handique and Chakraborty, 2015) by considering three data fitting
examples with real life data sets.
Key words: Beta Generated family, Generalized Marshall-Olkin family, Exponentiated family, AIC,
BIC and Power weighted moments.
1. Introduction
Here we briefly introduce the Beta-G (Eugene et al. 2002 and Jones, 2004) and Generalized
Marshall-Olkin family (Jayakumar and Mathew, 2008) of distributions.
1.1 Some formulas and notations
Here first we list some formulas to be used in the subsequent sections of this article.
If T is a continuous random variable with pdf, )(tf and cdf ][)( tTPtF , then its
Survival function (sf): )(1][)( tFtTPtF ,
Hazard rate function (hrf): )(/)()( tFtfth ,
Reverse hazard rate function (rhrf): )(/)()( tFtftr ,
2
Cumulative hazard rate function (chrf): )]([log)( tFtH ,
th),,( rqp Power Weighted Moment (PWM): dttftFtFt rqprqp )(])(1[])([,,
,
Rényi entropy:
dttfI R )(log)1()( 1 .
1.2 Beta-G family of distributions
The cdf of beta-G (Eugene et al. 2002 and Jones 2004) family of distribution is
)(tF BG )(
11 )1(),(
1 tF
o
nm vdvvnmB ),(
),()(
nmBnmB tF ),()( nmI tF (1)
Where t
nmt dxxxnmBnmI
0
111 )1(),(),( denotes the incomplete beta function ratio.
The pdf corresponding to (1) is
)(tf BG 11 )](1[)()(),(
1 nm tFtFtfnmB
11 )()()(),(
1 nm tFtFtfnmB
(2)
Where dttFdtf )()( .
sf: ),(1)( )( nmItF tFBG
),(),(),( )(
nmBnmBnmB tF
hrf: )()()( tFtfth BGBGBG ),(),(
)()()()(
11
nmBnmBtFtFtf
tF
nm
rhrf: )()()( tFtftr BGBGBG ),(
)()()()(
11
nmBtFtFtf
tF
nm
chrf: )(tH
),(),(),(
log )(
nmBnmBnmB tF
Some of the well known beta generated families are the Beta-generated (beta-G) family (Eugene
et al., 2002; Jones 2004), beta extended G family (Cordeiro et al., 2012), Kumaraswamy beta
generalized family (Pescim et al., 2012), beta generalized weibull distribution (Singla et al., 2012),
beta generalized Rayleigh distribution (Cordeiro et al., 2013), beta extended half normal
distribution (Cordeiro et al., 2014), beta log-logistic distribution (Lemonte, 2014) beta generalized
inverse Weibull distribution (Baharith et al., 2014), beta Marshall-Olkin family of distribution
(Alizadeh et al., 2015) and beta generated Kumaraswamy-G family of distribution (Handique and
Chakraborty 2016a) among others.
1.3 Generalized Marshall-Olkin Extended ( GMOE ) family of distributions
Jayakumar and Mathew (2008) proposed a generalization of the Marshall and Olkin (1997) family of
distributions by using the Lehman second alternative (Lehmann 1953) to obtain the sf )(tF GMO of
the GMOE family of distributions by exponentiation the sf of MOE family of distributions as
3
)(tF GMO
0;0;,)(1
)( ttG
tG (3)
where 0, t ( 1 ) and 0 is an additional shape parameter. When ,1
)()( tFtF MOGMO and for ,1 )()( tFtF GMO . The cdf and pdf of the GMOE distribution
are respectively
)(tF GMO
)(1
)(1tG
tG (4)
and )(tf GMO
2
1
)](1[)(
)(1)(
tGtg
tGtG
1
1
)](1[)()(
tGtGtg (5)
Reliability measures like the hrf, rhrf and chrf associated with (1) are
)()()(
tFtfth GMO
GMOGMO
)(1)(
)](1[)()(
1
1
tGtG
tGtGtg
)(1
1)()(
tGtGtg
)(1)(
tGth
)()()(
tFtftr GMO
GMOGMO
)(1)(1
)](1[)()(
1
1
tGtG
tGtGtg
])()}(1[{)](1[)()( 1
tGtGtGtGtg
)](1[)()](1[)()(
1
1
tGtGtGtGtg
)(tH GMO
)(1
)(log)(1
)(logtG
tGtG
tG
Where )(tg , )(tG , )(tG and )(th are respectively the pdf, cdf, sf and hrf of the baseline distribution.
We denote the family of distribution with pdf (3) as ),,,(GMOE ba which for 1 , reduces
to ),,(MOE ba .
Some of the notable distributions derived Marshall-Olkin Extended exponential distribution
(Marshall and Olkin, 1997), Marshall-Olkin Extended uniform distribution (Krishna, 2011; Jose and
Krishna 2011), Marshall-Olkin Extended power log normal distribution (Gui, 2013a), Marshall-
Olkin Extended log logistic distribution (Gui, 2013b), Marshall-Olkin Extended Esscher transformed
Laplace distribution (George and George, 2013), Marshall-Olkin Kumaraswamy-G familuy of
distribution (Handique and Chakraborty, 2015a), Generalized Marshall-Olkin Kumaraswamy-G
family of distribution (Handique and Chakraborty 2015b) and Kumaraswamy Generalized Marshall-
Olkin family of distribution (Handique and Chakraborty 2016b).
4
In this article we propose a family of Beta generated distribution by considering the
Generalized Marshall-Olkin family (Jayakumar and Mathew, 2008) as the base line distribution in
the Beta-G family (Eugene et al. 2002 and Jones, 2004). This new family referred to as the Beta
Generalized Marshall-Olkin family of distribution is investigated for some its general properties.
The rest of this article is organized in seven sections. In section 2 the new family is introduced along
with its physical basis. Important special cases of the family along with their shape and main
reliability characteristics are presented in the next section. In section 4 we discuss some general
results of the proposed family. Different methods of estimation of parameters along with three
comparative data modelling applications are presented in section 5. The article ends with a
conclusion in section 6 followed by an appendix to derive asymptotic confidence bounds.
2. New Generalization: Beta Generalized Marshall-Olkin-G ( GBGMO ) family of
distributions
Here we propose a new Beta extended family by considering the cdf and pdf of GMO (Jayakumar
and Mathew, 2008) distribution in (4) and (5) as the )(and)( tFtf respectively in the Beta
formulation in (2) and call it GBGMO distribution. The pdf of GBGMO is given by
)(tf BGMOG
11
1
1
)(1)(
)(1)(1
)](1[)()(
),(1
nm
tGtG
tGtG
tGtGtg
nmB
(6)
, 0,0,,0,0 nmbat
Similarly substituting from equation (4) in (1) we get the cdf of GBGMO respectively as
)(tF BGMOG ),()(1
)(1nmI
tGtG
(7)
The sf, hrf, rhrf and chrf of GBGMO distribution are respectively given by
sf: )(tF BGMOG ),(1)(1
)(1nmI
tGtG
hrf :
)8()(1
)()(1
)(1
]),(1[)](1[)()(
),(1)(
11
)(1)(
1
1
1
nm
tGtG
BGMOG
tGtG
tGtG
nmItGtGtg
nmBth
rhrf :
5
)9()(1
)()(1
)(1
]),([)](1[)()(
),(1)(
11
)(1)(1
1
1
nm
tGtG
BGMOG
tGtG
tGtG
nmItGtGtg
nmBtr
chrf: ]),(1[log)()(1
)(1nmItH
tGtG
BGMOG
Remark: The ),n,(m,BGMO G reduces to
(i) )n,(m,BMO (Alizadeh et al., 2015) for 1 ; (ii) ),(GMO ,(Jayakumar and
Mathew, 2008) if 1 nm ; (iii) )(MO (Marshall and Olkin, 1997) when
1 nm ; and (iv) n)(m,B (Eugene et al., 2002; Jones 2004) for 1 .
2.1 Genesis of the distribution
If m and n are both integers, then the probability distribution takes the same form as the order
statistics of the random variable T .
Proof: Let 121 ...,, nmTTT be a sequence of i.i.d. random variables with cdf batG ])(1[1 . Then
the pdf of the thm order statistics )(mT is given by
11
)1(1
)](1[)()(
]})(1)([{]})(1)({1[!])1[(!)1(
!)1(
tGtGtg
tGtGtGtGmnmm
nm mnmm
11
11
)](1[)()(
]})(1)([{]})(1)({1[)()(
)(
tGtGtg
tGtGtGtGnm
nm nm
11
11
)](1[)()(
]})(1)([{]})(1)({1[),(
1
tGtGtg
tGtGtGtGnmB
nm
2.2 Shape of the density function
Here we have plotted the pdf of the GBGMO for some choices of the distribution G and parameter
values to study the variety of shapes assumed by the family.
6
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
BGMO-E
X
Den
sity
m=0.6,n=2,=2,=0.5,=0.9m=1.3,n=1.9,=1.6,=0.6,=0.1m=3.5,n=2,=0.5,=1.6,=0.5m=19.5,n=1.9,=0.9,=1.6,=0.6m=19.5,n=1.9,=0.9,=1.6,=0.5m=2.9,n=2.2,=2.5,=0.5,=0.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.5
1.0
1.5
2.0
2.5
BGMO-W
X
Den
sity
m=1,n=2,=1.5,=2.1,=0.9,=0.6m=13,n=3.5,=2.5,=0.6,=0.5,=0.5m=50,n=8.5,=2.2,=0.89,=0.5,=0.5m=0.05,n=0.45,=1.4,=1.5,=0.5,=0.5m=0.2,n=0.4,=3.5,=1.5,=0.5,=0.4m=40,n=9.5,=2.5,=0.5,=0.5,=0.5
(a) (b)
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
BGMO-L
X
Den
sity
m=8.2,n=1.9,=5.55,=3.5,=0.48,=0.8m=7.2,n=1.9,=5.45,=3.1,=0.42,=0.7m=5,n=1.56,=2.5,=0.5,=0.5,=0.5m=5.2,n=1.56,=4.5,=2.5,=0.5,=0.5m=8.2,n=1.9,=5.55,=3.2,=0.45,=0.6m=1,n=3.5,=2.5,=0.5,=0.5,=0.5
0 1 2 3 4 5 6 7
0.0
0.5
1.0
1.5
2.0
2.5
BGMO-Fr
X
Den
sity
m=0.1,n=2.5,=1.4,=3.5,=0.5,=0.5m=0.3,n=2.1,=1.3,=4.5,=0.5,=0.5m=0.23,n=4.45,=1.3,=4.5,=0.5,=0.5m=0.26,n=3.35,=1.4,=4.5,=0.3,=0.5m=0.05,n=2.5,=1.2,=4.5,=0.5,=0.5m=0.2,n=2.5,=1.3,=4.5,=0.5,=0.5
(c) (d)
Fig 1: Density plots (a) EBGMO , (b) WBGMO , (c) LBGMO and (d) FrBGMO
Distributions:
7
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
BGMO-E
X
h(x)
m=0.05,n=1.9,=1.6,=0.9,=0.7m=23.5,n=1.99,=0.58,=1.6,=0.5m=0.04,n=1.99,=0.58,=1.6,=0.5m=19.5,n=1.84,=0.9,=2.8,=0.9m=15.5,n=1.9,=0.8,=1.6,=0.5m=0.89,n=2.78,=0.48,=1.5,=0.5
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
BGMO-W
X
h(x)
m=0.01,n=0.65,=1,=1.9,=0.9,=0.8m=0.08,n=0.3,=2.1,=1.83,=0.8,=0.7m=0.04,n=0.25,=1.7,=2,=0.8,=1.8m=0.05,n=0.45,=1.5,=1.8,=0.8,=0.8m=0.08,n=0.15,=2,=2.2,=0.8,=1.8m=0.02,n=0.65,=0.9,=1.9,=0.9,=0.8
(a) (b)
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
BGMO-L
X
h(x)
m=2,n=0.35,=2.5,=5.5,=2.9,=1.9m=7.2,n=1.9,=5.45,=3.1,=0.42,=0.7m=5,n=1.56,=2.5,=0.5,=0.5,=0.5m=4,n=1.56,=4.5,=2.5,=0.5,=0.5m=4.2,n=1.99,=6.55,=9.2,=0.45,=0.6m=0.5,n=3.8,=0.19,=3.5,=0.6,=0.9
0 1 2 3 4 5 6 7
0.0
0.5
1.0
1.5
2.0
2.5
BGMO-Fr
X
h(x)
m=0.6,n=1.9,=0.25,=5.5,=0.8,=0.9m=0.99,n=2.5,=3,=3.5,=0.5,=0.5m=5.23,n=4.65,=2.3,=6.7,=0.5,=0.5m=1.9,n=2.5,=3,=3.5,=0.63,=0.7m=0.3,n=1.8,=0.2,=3.5,=0.5,=0.8m=7.23,n=4.95,=2.3,=6.7,=0.41,=0.5
(c) (d) Fig 2: Hazard plots plots (a) EBGMO , (b) WBGMO , (c) LBGMO and (d) FrBGMO
Distributions:
From the plots in figure 1 and 2 it can be seen that the family is very flexible and can offer
many different types of shapes of density and hazard rate function including the bath tub shaped free
hazard.
3. Some special GBGMO distributions
Some special cases of the GBGMO family of distributions are presented in this section.
3.1 The BGMO exponential ( EBGMO ) distribution
Let the base line distribution be exponential with parameter ,0 0,):( tetg t
and ,1):( tetG 0t then for the EBGMO model we get the pdf and cdf respectively as:
8
)(tf BGMOE
11
1
1
111
]1[][
),(1
n
t
tm
t
t
t
tt
ee
ee
eee
nmB
and )(tF BGMOE ),(1
1
nmIt
t
ee
sf: )(tF BGMOE ),(11
1
nmIt
t
ee
hrf :
11
11
1
1
111
]),(1[]1[][
),(1)(
n
t
tm
t
t
ee
t
ttBGMOE
ee
ee
nmIeee
nmBth
t
t
rhrf :
11
11
1
1
111
]),([]1[][
),(1)(
n
t
tm
t
t
ee
t
ttBGMOE
ee
ee
nmIeee
nmBtr
t
t
chrf: ]),(1[log)(1
1
nmItHt
t
ee
BGMOE
3.2 The BGMO Lomax ( LBGMO ) distribution
Considering the Lomax distribution (Ghitany et al. 2007) with pdf and cdf given
by 0,)](1[)(),:( )1( tttg , and ,])(1[1),:( ttG 0 and 0 the
pdf and cdf of the LBGMO distribution are given by
11
1
1)1(
])(1[1])(1[
])(1[1])(1[1
]])(1[1[]])(1[[)](1[)(
),(1)(
nm
BGMOL
tt
tt
ttt
nmBtf
and )(tF BGMOL ),(])(1[1
])(1[1
nmIt
t
sf: )(tF BGMOL ),(1)](1[1
)](1[1nmI
tt
9
hrf : )(thBGMOL
11
)](1[1)](1[
1
1
)1()1(
)](1[1)](1[
)](1[1)](1[1
),(11
])}(1{1[)](1[)](1[)(
),(1
nm
tt
tt
tt
nmIttt
nmB
rhrf: )(tr BGMOL
11
)](1[1)](1[
1
1
)1()1(
)](1[1)](1[
)](1[1)](1[1
),(1
])}(1{1[)](1[)](1[)(
),(1
nm
tt
tt
tt
nmIttt
nmB
chrf: ]),(1[log)()](1[1
)](1[1
nmItHt
t
BGMOL
3.3 The BGMO Weibull ( WBGMO ) distribution
Considering the Weibull distribution (Ghitany et al. 2005, Zhang and Xie 2007) with parameters
0 and 0 having pdf and cdf tettg 1)( and
tetG 1)( respectively we get
the pdf and cdf of WBGMO distribution as
)(tf BGMOW
11
1
11
111
]1[][
),(1
n
t
tm
t
t
t
tt
ee
ee
eeet
nmB
and )(tF BGMOW ),(
11
nmIt
t
e
e
sf: )(tF BGMOW ),(1
11
nmIt
t
e
e
hrf: )(thBGMOW
11
11
1
11
111
),(11
]1[][
),(1
n
t
tm
t
t
e
e
t
tt
ee
ee
nmIeeet
nmBt
t
10
rhrf:
11
11
1
11
111
),(1
]1[][
),(1)(
n
t
tm
t
t
e
e
t
ttBGMOW
ee
ee
nmIeeet
nmBtr
t
t
chrf: ]),(1[log)(
11
nmItHt
t
e
e
BGMOW
3.4 The BGMO Frechet ( FrBGMO ) distribution
Suppose the base line distribution is the Frechet distribution (Krishna et al., 2013) with pdf and cdf
given by )()1()( tettg and 0,)( )( tetG t respectively, and then the
corresponding pdf and cdf of FrBGMO distribution becomes
1
)(
)(1
)(
)(
1)(
1)()()1(
]1[1]1[
]1[1]1[1
]]1[1[]1[
),(1)(
n
t
tm
t
t
t
ttBGMOFr
ee
ee
eeet
nmBtf
and )(tF BGMOFr ),(
]1[1
]1[1
)(
)(nmI
t
t
e
e
sf: )(tF BGMOFr ),(1
]1[1
]1[1
)(
)(nmI
t
t
e
e
hrf : )(thBGMOFr
1
)(
)(1
)(
)(
]1[1
]1[1
1)(
1)()()1(
]1[1]1[
]1[1]1[1
),(11
]]1[1[]1[
),(1
)(
)(
n
t
tm
t
t
e
e
t
tt
ee
ee
nmIeeet
nmBt
t
rhrf : )(trBGMOFr
11
1
)(
)(1
)(
)(
]1[1
]1[1
1)(
1)()()1(
]1[1]1[
]1[1]1[1
),(1
]]1[1[]1[
),(1
)(
)(
n
t
tm
t
t
e
e
t
tt
ee
ee
nmIeeet
nmBt
t
chrf: ]),(1[log)(
]1[1
]1[1
)(
)(nmItH
t
t
e
e
BGMOFr
3.5 The BGMO Gompertz ( GoBGMO ) distribution
Next by taking the Gompertz distribution (Gieser et al. 1998) with pdf and cdf )1(
)(
tet eetg
and 0,0,0,1)()1(
tetGte
respectively, we get the pdf and cdf of
GoBGMO distribution as
)(tf BGMOGo
1
)1(
)1(1
)1(
)1(
1)1(
1)1()1(
111
]1[
][),(
1
n
e
em
e
e
e
eet
t
t
t
t
t
tt
e
e
e
e
e
eeenmB
and )(tF BGMOGo ),(
)1(
)1(
1
1
nmI
te
te
e
e
sf: )(tF BGMOGo ),(1
)1(
)1(
1
1
nmI
te
te
e
e
hrf: )(thBGMOGo
1
)1(
)1(
1
)1(
)1(
1
1
1)1(
1)1()1(
111
),(11
]1[
][),(
1
)1(
)1(
n
e
em
e
e
e
e
e
eet
t
t
t
t
te
te
t
tt
e
e
e
e
nmIe
eeenmB
rhrf: )(tr BGMOGo
12
1
)1(
)1(1
)1(
)1(
1
1
1)1(
1)1()1(
111
),(1
]1[
][),(
1
)1(
)1(
n
e
em
e
e
e
e
e
eet
t
t
t
t
te
te
t
tt
e
e
e
e
nmIe
eeenmB
chrf: ]),(1[log)(
)1(
)1(
1
1
nmItH
te
te
e
e
BGMOGo
3.6 The BGMO Extended Weibull ( EWBGMO ) distribution
The pdf and the cdf of the extended Weibull (EW) distributions of Gurvich et al. (1997) is given by
),:( tg ):()]:(exp[ tztZ and ),:( tG )]:(exp[1 tZ , 0, RDt
where ):( tZ is a non-negative monotonically increasing function which depends on the parameter
vector . and ):( tz is the derivative of ):( tZ .
By considering EW as the base line distribution we derive pdf and cdf of the EWBGMO as
11
1
1
)]:(exp[1)]:(exp[
)]:(exp[1)]:(exp[1
}]):(exp{1[])}:(exp{[):()]:(exp[
),(1)(
nm
BGMOEW
tZtZ
tZtZ
tZtZtztZ
nmBtf
and )(tF BGMOEW ),()]:(exp[1
)]:(exp[1
nmItZ
tZ
Important models can be seen as particular cases with different choices of ):( tZ :
(i) ):( tZ = t : exponential distribution.
(ii) ):( tZ = 2t : Rayleigh (Burr type-X) distribution.
(iii) ):( tZ = )(log kt : Pareto distribution
(iv) ):( tZ = ]1)([exp1 t : Gompertz distribution.
sf: )(tF BGMOEW ),(1)]:(exp[1
)]:(exp[1
nmItZ
tZ
13
hrf: )(thBGMOEW
11
)]:(exp[1)]:(exp[1
1
1
)]:(exp[1)]:(exp[
)]:(exp[1)]:(exp[1
),(11
}]):(exp{1[])}:(exp{[):()]:(exp[
),(1
nm
tZtZ
tZtZ
tZtZ
nmItZtZtztZ
nmB
rhrf: )(tr BGMOEW
11
)]:(exp[1)]:(exp[1
1
1
)]:(exp[1)]:(exp[
)]:(exp[1)]:(exp[1
),(11
}]):(exp{1[])}:(exp{[):()]:(exp[
),(1
nm
tZtZ
tZtZ
tZtZ
nmItZtZtztZ
nmB
chrf: ]),(1[log)()]:(exp[1
)]:(exp[1
nmItHtZ
tZBGMOEW
3.7 The BGMO Extended Modified Weibull ( EMWBGMO ) distribution
The modified Weibull (MW) distribution (Sarhan and Zaindin 2013) with cdf and pdf is given by
),,;( tG ][exp1 tt , 0,0,,0,0 t and
),,;( tg ][exp)( 1 ttt respectively.
The corresponding pdf and cdf of EMWBGMO are given by
11
1
11
][exp1][exp
][exp1][exp1
}]]{exp1[]}{exp[][exp)(
),(1)(
nm
BGMOEMW
tttt
tttt
ttttttt
nmBtf
and )(tF BGMOEMW ),(][exp1
][exp1
nmItt
tt
sf: )(tF BGMOEMW ),(1][exp1
][exp1
nmItt
tt
hrf : )(thBGMOEMW
14
11
][exp1][exp
1
1
11
][exp1][exp
][exp1][exp1
),(11
}]]{exp1[]}{exp[][exp)(
),(1
nm
tttt
tttt
tttt
nmIttttttt
nmB
rhrf: )(tr BGMOEMW
11
][exp1][exp1
1
11
][exp1][exp
][exp1][exp1
),(1
}]]{exp1[]}{exp[][exp)(
),(1
nm
tttt
tttt
tttt
nmIttttttt
nmB
chrf: ]),(1[log)(][exp1
][exp1
nmItHtt
tt
BGMOEMW
3.8 The BGMO Extended Exponentiated Pareto ( EEPBGMO ) distribution
The pdf and cdf of the exponentiated Pareto distribution, of Nadarajah (2005), are given respectively
by 1)1( ])(1[)( kkk ttktg and ])(1[)( kttG , x and 0,, k . Thus the
pdf and the cdf of EEPBGMO distribution are given by
11
1
11)1(
]})(1{1[1]})(1{1[
]})(1{1[1]})(1{1[1
]]})(1{1[1[]})(1{1[])(1[
),(1)(
n
k
km
k
k
k
kkkkBGMOEEP
tt
tt
ttttk
nmBtf
and )(tF BGMOEEP ),(]})(1{1[1
]})(1{1[1
nmIk
k
tt
sf: )(tF BGMOEEP ),(1]})(1{1[1
]})(1{1[1
nmIk
k
tt
hrf: )(thBGMOEEP
11
]})(1{1[1]})(1{1[
1
1
11)1(
]})(1{1[1]})(1{1[
]})(1{1[1]})(1{1[1
),(11
]]})(1{1[1[]})(1{1[])(1[
),(1
n
k
km
k
k
tt
k
kkkk
tt
tt
nmIttttk
nmBk
k
15
rhrf: )(tr BGMOEEP
11
]})(1{1[1]})(1{1[1
1
11)1(
]})(1{1[1]})(1{1[
]})(1{1[1]})(1{1[1
),(1
]]})(1{1[1[]})(1{1[])(1[
),(1
n
k
km
k
k
tt
k
kkkk
tt
tt
nmIttttk
nmBk
k
chrf: ]),(1[log)(]})(1{1[1
]})(1{1[1
nmItHk
k
tt
BGMOEEP
4. General results for the Beta Generalized Marshall-Olkin ( GBGMO ) family of
distributions
In this section we derive some general results for the proposed GBGMO family.
4.1 Expansions
By using binomial expansion in (6), we obtain
),,,;( nmtf BGMOG
11
1
1
)(1)(
)(1)(1
)](1[)()(
),(1
nm
tGtG
tGtG
tGtGtg
nmB
111
2 )(1)(
)(1)(1
)](1[)(
)](1[)(
),(
nm
tGtG
tGtG
tGtG
tGtg
nmB
)1(11 ]),([])},({1[)],([),(),(
nMOmMOMOMO tFtFtFtfnmB
)1(1 ])},({1[)],([),(),(
mMOnMOMO tFtFtfnmB
jMOjm
j
nMOMO tFj
mtFtf
nmB )];([)1(
1)],([),(
),(
1
0
1
1)(1
0)];([)1(
1),(
),(
njMO
m
j
jMO tFj
mtf
nmB
1)(1
0)];([),(
njMOm
jj
MO tFtf (10)
)(1
0)];([
)(njMO
m
j
j tFdtd
nj
16
)(1
0)];([ njMO
m
jj tFdtd
(11)
)))]((;([1
0njtF
dtd MO
m
jj
)))((;(1
0njtf MO
m
jj
(12)
Where
jm
njnmB
j
j
1)(),(
)1( 1
and )( njjj
Alternatively, we can expand the pdf as
1)(1
0)];([),(
njMOm
jj
MO tFtf
lMOnj
l
lm
jj
MO tFl
njtf ]);([)1(
1)(),(
1)(
0
1
0
lMOnj
ll
MO tFtf ]);([),(1)(
0
(13)
11)(
0]);([
1
lMOnj
l
l tFdtd
l
11)(
0]);([
lMO
nj
ll tFdtd
Where
lnj
j
ljl
1)()1(
0
and
lnj
l j
ljl
1)()1(
11
0
We can expand the cdf as (see “Incomplete Beta Function” From Math World—A Wolfram Web
Resource. http://mathworld. Wolfram.com/ Incomplete Beta Function. html)
)14()(
)1(1)(!)1(!
!)1()1(
symbol. Pochhammer a is(x)Where)(!
)1(),(),;(
0
0
n0
n
n
na
n
n
na
n
n
naz
znan
bz
znanbn
bz
znan
bzbaBbazB
)(tF BGMOG ),()(1
)(1nmI
tGtG
Using (14) in (7) we have
17
ii
i
m
tGtG
imin
tGtG
nmB
)(1)(1
)()1(1
)(1)(1
),(1
0
iMOi
i
mMO tFimi
ntF
nmB]}),({1[
)()1(1
]}),({1[),(
10
imMO
i
i
tFi
nimnmB
]}),({1[
1)(),(
)1(0
jMOj
ji
i
tFj
imi
nimnmB
]),([)1(1
)(),()1(
00
kMOj
k
k
ji
ji
tFk
jj
imi
nimnmB
]),([)1(1
)(),()1(
00,
kMO
ji
kjij
ktF
kj
jim
in
imnmB]),([
1)(),(
)1(0, 0
By exchanging the indices j and k in the sum symbol, we have
),,,;( nmtF BGMO kMO
ki
kji
kjtF
kj
jim
in
imnmB]),([
1)(),(
)1(0,
and then
),,,;( nmtF BGMO kMO
kk tF ),(
0
(15)
Where
kj
jim
in
imnmBi kj
kji
k
0
1)(),(
)1(
Similarly an expansion for the cdf of GBGMO can be derives as
),,,;( nmtF BGMOG ),(]),([1
nmItF MO
pnmMOpMOnm
mptFtF
pnm
1
1
]),([]}),({1[1
pnmMOqMOqp
q
nm
mptFtF
qp
pnm
1
0
1
]),([}),({)1(1
)1(1
0)],([
1)1( qpnmMO
nm
mp
p
q
q tFpnm
qp
rMOrqpnm
r
nm
mp
p
q
q tFr
qpnmpnm
qp
)],([)1()1(1
)1()1(
0
1
0
rMOnm
mp
p
q
rqqpnm
rtF
rqpnm
pnm
qp
)],([)1(1
)1(1
0
)1(
0
18
rMOnm
mp
p
qrqp
qpnm
rtF )],([
1
0,,
)1(
0
(16)
Where
rqpnm
pnm
qprq
rqp
)1(1)1(,,
4.2 Order statistics
Suppose nTTT ...,, 21 is a random sample from any GBGMO distribution. Let nrT : denote the thr
order statistics. The pdf of nrT : can be expressed as
)(: tf nrrnBGMOGrBGMOGBGMOG tFtFtf
rnrn
)}(1{)()(
!)(!)1(! 1
1
0)()()1(
!)(!)1(!
rjBGMOGBGMOG
rn
j
j tFtfj
rnrnr
n
Now using the general expansion of the GBGMO distribution pdf and cdf we get the pdf of the thr order statistics for of the GBGMO is given by
1
0
1)(
00:
),(
]);([),()1(!)(!)1(
!)(
rj
kMO
kk
lMOnj
ll
MOrn
j
jnr
tF
tFtfj
rnrnr
ntf
Where kl and defined in above
Now
0,1
1
0]),([),(
k
kMOkrj
rjkMO
kk tFdtF
Where
k
cckrjckrj dkrjc
kd
1,1
0,1 ])([1
(Nadarajah et. al 2015)
Therefore the density function of the thr order statistics of GBGMO distribution can be
expressed as
0,1
1)(
00:
]),([
]);([),()1(!)(!)1(
!)(
k
kMOkrj
lMOnj
ll
MOrn
j
jnr
tFd
tFtfj
rnrnr
ntf
lk
k
MOkrjl
nj
l
MOrn
j
j tFdtfj
rnrnr
n0
,1
1)(
00]),([),()1(
!)(!)1(!
lk
k
MOkl
nj
l
MO tFtf
0,
1)(
0
]),([),(
(17)
19
Where krjl
rn
j
jkl d
jrn
rnrn
,10
, )1(!)(!)1(
!
and krjl d ,1and defined above.
4.3 Probability weighted moments
The probability weighted moments (PWMs), first proposed by Greenwood et al. (1979), are
expectations of certain functions of a random variable whose mean exists. The thrqp ),,( PWM of T
is defined by
dttftFtFt rqprqp )()](1[)(,,
From equations (10) and (13) the ths moment of T can be written either as
dtnmtftTE BGMOGss ),,,;()(
dttftFt MOnjMOsm
jj ),()];([ 1)(
1
0
dttGtgtGtGt njsm
jj ])](1[)([)](1)([ 21)(
1
0
1)(,0,
1
0
njs
m
jj
or dttftFtTE MOlMOsnj
ll
s ),(]);([)(1)(
0
dttGtgtGtGt lsnj
ll ])](1[)([)](1)([ 2
1)(
0
0,,
1)(
0ls
nj
ll
Where dttGtgtGtGtGtGt rqprqp ])](1[)([)](1)([)](1)([ 2
,,
is the PWM of )(M O distribution.
Therefore the moments of the ),,,;(G-BGMO nmt can be expresses in terms of the PWMs
of )(M O (Marshall and Olkin, 1997). The PWM method can generally be used for estimating
parameters quantiles of generalized distributions. These moments have low variance and no severe
biases, and they compare favourably with estimators obtained by maximum likelihood.
20
Proceeding as above we can derive ths moment of the thr order statistic ,:nrT in a random
sample of size n from GBGMO on using equation (17) as
)( ;nrsTE
0
0,,,
1)(
0 klkskl
nj
l
Where j , l and kl , defined in above
4.5 Moment generating function
The moment generating function of GBGMO family can be easily expressed in terms of those of
the exponentiated MO (Marshall and Olkin, 1997) distribution using the results of section 4.1. For
example using equation (11) it can be seen that
dttFdtdedtnmtfeeEsM
m
j
njMOj
stBGMOstsTT
1
0
)()];([),,,;(][)(
)()];([1
0
)(1
0sMdttF
dtde X
m
jj
njMOstm
jj
Where )(sM X is the mgf of a MO (Marshall and Olkin, 1997) distribution.
4.6 Renyi Entropy
The entropy of a random variable is a measure of uncertainty variation and has been used in various
situations in science and engineering. The Rényi entropy is defined by
dttfI R )(log)1()( 1
where 0 and 1 For furthers details, see Song (2001). Using binomial expansion in (6) we
can write ),,,;( nmtf BGMOG
111
2 )(1)(
)(1)(1
)](1[)(
)](1[)(
),(
nm
tGtG
tGtG
tGtG
tGtg
nmB
)1()1()1( ]),([])},({1[)],([),(),(
nMOmMOMOMO tFtFtFtfnmB
)1()1( ])},({1[)],([),(),(
mMOnMOMO tFtFtfnmB
jMOjm
j
nMOMO tFj
mtFtf
nmB
)];([)1()1(
)],([),(),(
)1(
0
)1(
)1()1(
0)];([)1(
)1(),(
),(
njMOj
m
j
MO tFj
mtf
nmB
21
Thus the Rényi entropy of T can be obtained as
dttFtfZI njMOMOm
jjR
)1()1(
0
1 )];([),(log)1()(
Where jj j
mnmB
Z )1()1(
),(
4.7 Quantile power series and random sample generation
The quantile function T, let )()( 1 uFuQt , can be obtained by inverting (7). Let )(, uQz nm be
the beta quantile function. Then,
])}(1{1[1
])}(1{1[)( 1
1
,
,
uQ
uQQuQt
nm
nmG
It is possible to obtain an expansion for )(, uQ nm in the Wolfram website as
It is possible to obtain an expansion for )(, uQ nm as
0
, )(i
miinm ueuQz
(see “Power series” From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/PowerSeries.html)
Where im
i dnmBme 1]),([ and )1()1(,1,0 210 mnddd ,
)2()1(2
)453()1(2
2
3
mm
nmmnmnd
..].)3()2()1(3[]18)4731(
)43033()58()2()16([)1(3
22344
mmmnn
mnnmnnmnmnd
The Bowley skewness (Kenney and Keeping 1962) measures and Moors kurtosis (Moors
1988) measure are robust and less sensitive to outliers and exist even for distributions without
moments. For GBGMO family these measures are given by
)41()43()21(2)41()43(
QQQQQB
and )82()86(
)85()87()81()83(QQ
QQQQM
For example, let the G be exponential distribution with parameter ,0 having pdf and cdf as
0,):( tetg t and ,1):( tetG respectively. Then the thp quantile is obtained as
]1[log)/1( p . Therefore, the thp quantile pt , of EBGMO is given by
22
])}(1{1[1
])}(1{1[1log1
1
1
,
,
pQ
pQt
nm
nmp
4.8 Asymptotes
Here we investigate the asymptotic shapes of the proposed family following the method followed in
Alizadeh et al., (2015).
Proposition 1. The asymptotes of equations (6), (7) and (8) as 0t are given by
1]}1{1[),()(~)( m
nmBtgtf
as 0)( tG
m
mnmBtF ]}1{1[
),(1~)( as 0)( tG
1]}1{1[),()(~)( m
nmBtgth
as 0)( tG
Proposition 2. The asymptotes of equations (6), (7) and (8) as t are given by
),(
)()(~)(1
nmBtGtgtf
nn as t
),(
)]([~)(1nmBn
tGtFn
as t
1)()(~)( tGtgth as t
5. Estimation
5.1 Maximum likelihood method
The model parameters of the GBGMO distribution can be estimated by maximum likelihood. Let T
ntttt ),...,( 21 be a random sample of size n from GBGMO with parameter vector TTnm ),,,,( βθ , where T
q ),...,( 21 β corresponds to the parameter vector of the baseline distribution G. Then the log-likelihood function for θ is given by
)(θ )],(log[)],([log)1(]),([logloglog00
nmBrtGtgrrr
ii
r
ii
ββ
)18(]]}),(1{),([[log)1(
]]}),(1{),([1[log)1()],(1[log)1(
0
00
r
iii
r
iii
r
ii
tGtGn
tGtGmtG
ββ
βββ
This log-likelihood function can not be solved analytically because of its complex form but it can be
maximized numerically by employing global optimization methods available with software’s like R,
SAS, Mathematica or by solving the nonlinear likelihood equations obtained by differentiating (18).
23
By taking the partial derivatives of the log-likelihood function with respect to
,,,nm and β components of the score vector
Tnm TUUUUUU ),,,,(
θ can be obtained as follows:
r
iiim tGtGnmrmr
mU
0]]}),(1{),([1[log)()( ββ
r
iiin tGtGbnmrnr
nU
0]]}),(1{),([[log)()( ββ
U
r
ii
r
ii tGtGrr
00)],(1[log)],([loglog ββ
r
i ii
iiii
r
i ii
iiii
tGtGtGtGtGtGn
tGtGtGtGtGtGm
0
0
]}),(1{),([]}),(1{),([log]}),(1{),([)1(
]}),(1{),([1}]),(1{),([log]}),(1{),([)1(
ββββββ
ββββββ
U
r
i i
i
tGtGr
0 ),(1),()1(β
β
r
i iii
ii
tGtGtGtGtGm
0 )],(1[])},({)},(1{[),(),()1(
βββββ
r
i i
i
tGtGn
0 ),(1),()1(1β
β
β
U
r
i i
ir
i i
i
tGtG
tgtg
0
)(
0
)(
),(),()1(
),(),(
ββ
ββ ββ
r
i i
i
tGtG
0
)(
),(1),()1(βββ
r
i iii
ii
tGtGtGtGtGm
0
)(1
)],(1[])},({)},(1{[),(),()1(
βββββ β
r
i ii
i
tGtGtGn
0
)(
),()],(1[),()1(
ββββ
Where (.) is the digamma function.
5.2 Asymptotic standard error and confidence interval for the mles:
The asymptotic variance-covariance matrix of the MLEs of parameters can obtained by inverting the
Fisher information matrix )(I θ which can be derived using the second partial derivatives of the log-
likelihood function with respect to each parameter. The thji elements of )(I θn are given by
qjilEji
ji
3,,2,1,,)(I2
θ
The exact evaluation of the above expectations may be cumbersome. In practice one can estimate
)(I θn by the observed Fisher’s information matrix )ˆ(I θn is defined as:
24
qjil
jiji
3,,2,1,,)(Iˆ
2
θθ
θ
Using the general theory of MLEs under some regularity conditions on the parameters as n the
asymptotic distribution of )ˆ( θθ n is ),0( nk VN where )(I)( 1 θ njjn vV . The asymptotic
behaviour remains valid if nV is replaced by )ˆ(Iˆ 1 θnV . This result can be used to provide large
sample standard errors and also construct confidence intervals for the model parameters. Thus an
approximate standard error and %100)2/1( confidence interval for the mle of jth parameter j are
respectively given by jjv and jjj vZ ˆˆ2/ , where 2/Z is the 2/ point of standard normal
distribution.
As an illustration on the MLE method its large sample standard errors, confidence interval in
the case of ),,,,( nmEBGMO is discussed in an appendix.
5.3 Real life applications
In this subsection, we consider fitting of three real data sets to show that the proposed GBGMO
distribution can be a better model than GGMOKw (Handique and Chakraborty, 2015) by taking
as Weibull distribution as G. We have estimated the parameters by numerical maximization of
loglikelihood function and provided their standard errors and 95% confidence intervals using large
sample approach (see appendix).
In order to compare the distributions, we have considered known criteria like AIC (Akaike
Information Criterion), BIC (Bayesian Information Criterion), CAIC (Consistent Akaike Information
Criterion) and HQIC (Hannan-Quinn Information Criterion). It may be noted that lkAIC 22 ;
lnkBIC 2)(log ; 1)1(2
knkkAICCAIC ; and lnkHQIC 2)]([loglog2
where k is the number of parameters in the statistical model, n the sample size and l is the
maximized value of the log-likelihood function under the considered model. In these applications
method of maximum likelihood will be used to obtain the estimate of parameters.
Example I:
The following data set gives the time to failure )10( 3 h of turbocharger of one type of engine given in
Xu et al. (2003).
{1.6, 2.0, 2.6, 3.0, 3.5, 3.9, 4.5, 4.6, 4.8, 5.0, 5.1, 5.3, 5.4, 5.6, 5.8, 6.0, 6.0, 6.1, 6.3, 6.5, 6.5, 6.7, 7.0,
7.1, 7.3, 7.3, 7.3, 7.7, 7.7, 7.8, 7.9, 8.0, 8.1, 8.3, 8.4, 8.4, 8.5, 8.7, 8.8, 9.0}
25
Table 1: MLEs, standard errors and 95% confidence intervals (in parentheses) and
the AIC, BIC, CAIC and HQIC values for the data set.
Parameters WGMOKw WBGMO
a
1.178 (0.017)
(1.14, 1.21)
1.187 (0.702)
(-0.19, 2.56)
b 0.291
(0.209) (-0.12, 0.70)
2.057 (2.240)
(-2.33, 6.45)
0.617 (0.002)
(0.61, 0.62)
0.009 (0.006)
(-0.00276, 0.02)
1.855 (0.003)
(1.85, 1.86)
4.194 (0.668)
(2.88, 5.50)
1.619 (0.977)
(-0.29, 3.53)
0.047 (0.108)
(-0.16, 0.26)
0.178 (0.144)
(-0.10, 0.46)
0.017 (0.016)
(-0.01, 0.05) log-likelihood( maxl ) -90.99 -80.38
AIC 193.98 172.76 BIC 204.11 182.89
CAIC 196.53 175.31 HQIC 197.65 176.43
Estimated pdf's
X
Den
sity
0 2 4 6 8 10
0.00
0.05
0.10
0.15
0.20
0.25
BGMO-W GMOKw-W
0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0
Estimated cdf's
X
cdf
BGMO-W GMOKw-W
(a) (b)
Fig: 3 Plots of the observed (a) histogram and estimated pdf’s and (b) estimated cdf’s for the
GBGMO and GGMOKw for example I.
26
Example II:
The following data is about 346 nicotine measurements made from several brands of cigarettes in
1998. The data have been collected by the Federal Trade Commission which is an independent
agency of the US government, whose main mission is the promotion of consumer protection. [http:
//www.ftc.gov/ reports/tobacco or http: // pw1.netcom.com/ rdavis2/ smoke. html.]
{1.3, 1.0, 1.2, 0.9, 1.1, 0.8, 0.5, 1.0, 0.7, 0.5, 1.7, 1.1, 0.8, 0.5, 1.2, 0.8, 1.1, 0.9, 1.2, 0.9, 0.8, 0.6, 0.3,
0.8, 0.6, 0.4, 1.1, 1.1, 0.2, 0.8, 0.5, 1.1, 0.1, 0.8, 1.7, 1.0, 0.8, 1.0, 0.8, 1.0, 0.2, 0.8, 0.4, 1.0, 0.2, 0.8,
1.4, 0.8, 0.5, 1.1, 0.9, 1.3, 0.9, 0.4, 1.4, 0.9, 0.5, 1.7, 0.9, 0.8, 0.8, 1.2, 0.9, 0.8, 0.5, 1.0, 0.6, 0.1, 0.2,
0.5, 0.1, 0.1, 0.9, 0.6, 0.9, 0.6, 1.2, 1.5, 1.1, 1.4, 1.2, 1.7, 1.4, 1.0, 0.7, 0.4, 0.9, 0.7, 0.8, 0.7, 0.4, 0.9,
0.6, 0.4, 1.2, 2.0, 0.7, 0.5, 0.9, 0.5, 0.9, 0.7, 0.9, 0.7, 0.4, 1.0, 0.7, 0.9, 0.7, 0.5, 1.3, 0.9, 0.8, 1.0, 0.7,
0.7, 0.6, 0.8, 1.1, 0.9, 0.9, 0.8, 0.8, 0.7, 0.7, 0.4, 0.5, 0.4, 0.9, 0.9, 0.7, 1.0, 1.0, 0.7, 1.3, 1.0, 1.1, 1.1,
0.9, 1.1, 0.8, 1.0, 0.7, 1.6, 0.8, 0.6, 0.8, 0.6, 1.2, 0.9, 0.6, 0.8, 1.0, 0.5, 0.8, 1.0, 1.1, 0.8, 0.8, 0.5, 1.1,
0.8, 0.9, 1.1, 0.8, 1.2, 1.1, 1.2, 1.1, 1.2, 0.2, 0.5, 0.7, 0.2, 0.5, 0.6, 0.1, 0.4, 0.6, 0.2, 0.5, 1.1, 0.8, 0.6,
1.1, 0.9, 0.6, 0.3, 0.9, 0.8, 0.8, 0.6, 0.4, 1.2, 1.3, 1.0, 0.6, 1.2, 0.9, 1.2, 0.9, 0.5, 0.8, 1.0, 0.7, 0.9, 1.0,
0.1, 0.2, 0.1, 0.1, 1.1, 1.0, 1.1, 0.7, 1.1, 0.7, 1.8, 1.2, 0.9, 1.7, 1.2, 1.3, 1.2, 0.9, 0.7, 0.7, 1.2, 1.0, 0.9,
1.6, 0.8, 0.8, 1.1, 1.1, 0.8, 0.6, 1.0, 0.8, 1.1, 0.8, 0.5, 1.5, 1.1, 0.8, 0.6, 1.1, 0.8, 1.1, 0.8, 1.5, 1.1, 0.8,
0.4, 1.0, 0.8, 1.4, 0.9, 0.9, 1.0, 0.9, 1.3, 0.8, 1.0, 0.5, 1.0, 0.7, 0.5, 1.4, 1.2, 0.9, 1.1, 0.9, 1.1, 1.0, 0.9,
1.2, 0.9, 1.2, 0.9, 0.5, 0.9, 0.7, 0.3, 1.0, 0.6, 1.0, 0.9, 1.0, 1.1, 0.8, 0.5, 1.1, 0.8, 1.2, 0.8, 0.5, 1.5, 1.5,
1.0, 0.8, 1.0, 0.5, 1.7, 0.3, 0.6, 0.6, 0.4, 0.5, 0.5, 0.7, 0.4, 0.5, 0.8, 0.5, 1.3, 0.9, 1.3, 0.9, 0.5, 1.2, 0.9,
1.1, 0.9, 0.5, 0.7, 0.5, 1.1, 1.1, 0.5, 0.8, 0.6, 1.2, 0.8, 0.4, 1.3, 0.8, 0.5, 1.2, 0.7, 0.5, 0.9, 1.3, 0.8, 1.2,
0.9}
27
Table 2: MLEs, standard errors and 95% confidence intervals (in parentheses) and
the AIC, BIC, CAIC and HQIC values for the nicotine measurements data.
Parameters WGMOKw WBGMO
a
0.765 (0.025)
(0.72, 0.81)
0.866 (0.159)
(0.55, 1.18)
b 2.139
(0.774) (0.62, 3.66)
0.329 (0.167)
(0.00168, 0.66)
4.271 (0.018)
(4.24, 4.31)
2.131 (1.182)
(-0.19, 4.45)
2.919 (0.013)
(2.89, 2.94)
2.223 (0.725)
(0.80, 3.64)
1.097 (0.309)
(0.49, 1.70)
3.285 (3.901)
(-4.36, 10.93)
0.114 (0.042)
(0.03, 0.19)
2.635 (1.310)
(0.07, 5.20) log-likelihood( maxl ) -111.75 -109.28
AIC 235.50 230.56 BIC 258.58 253.64
CAIC 235.75 230.80 HQIC 244.69 239.76
Estimated pdf's
X
Den
sity
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
BGMO-W GMOKw-W
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Estimated cdf's
X
cdf
BGMO-W GMOKw-W
(a) (b)
Fig: 4 Plots of the observed (a) histogram and estimated pdf’s and (b) estimated cdf’s for the
GBGMO and GGMOKw for example II.
28
Example III:
This data set consists of 100 observations of breaking stress of carbon fibres (in Gba) given by
Nichols and Padgett (2006).
{3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.40, 3.15, 2.67,3.31, 2.81, 0.98, 5.56,
5.08, 0.39, 1.57, 3.19, 4.90, 2.93, 2.85, 2.77, 2.76, 1.73, 2.48, 3.68, 1.08, 3.22, 3.75, 3.22, 2.56, 2.17,
4.91, 1.59, 1.18, 2.48, 2.03, 1.69, 2.43, 3.39, 3.56, 2.83, 3.68, 2.00, 3.51, 0.85, 1.61, 3.28, 2.95, 2.81,
3.15, 1.92, 1.84, 1.22, 2.17, 1.61, 2.12, 3.09, 2.97, 4.20, 2.35, 1.41, 1.59, 1.12, 1.69, 2.79, 1.89, 1.87,
3.39, 3.33, 2.55, 3.68, 3.19, 1.71, 1.25, 4.70, 2.88, 2.96, 2.55, 2.59, 2.97, 1.57, 2.17, 4.38, 2.03, 2.82,
2.53, 3.31, 2.38, 1.36, 0.81, 1.17, 1.84, 1.80, 2.05, 3.65}.
Table 3: MLEs, standard errors and 95% confidence intervals (in parentheses) and
the AIC, BIC, CAIC and HQIC values for the breaking stress of carbon fibres data.
Parameters WGMOKw WBGMO
a
1.015 (0.071)
(0.88, 1.15)
1.458 (1.123)
(-0.74, 3.66)
b 0.385
(0.168) (0.06, 0.71)
0.734 (1.810)
(-2.81, 4.28)
0.803 (0.003)
(0.79, 0.81)
0.598 (1.496)
(-2.33, 3.53)
2.222 (0.004)
(2.21, 2.23)
2.439 (0.779)
(0.91, 3.97)
1.482 (0.440)
(0.62, 2.34)
0.685 (1.150)
(-1.57, 2.94)
0.345 (0.169)
(0.01, 0.68)
0.201 (0.573)
(-0.92, 1.32) log-likelihood( maxl ) 142.63 -141.29
AIC 297.26 294.58 BIC 312.89 310.21
CAIC 298.16 295.48 HQIC 303.59 300.92
29
Estimated pdf's
X
Den
sity
0 1 2 3 4 5 6
0.0
0.1
0.2
0.3
0.4
0.5
BGMO-W GMOKw-W
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
Estimated cdf's
X
cdf
BGMO-W GMOKw-W
(a) (b)
Fig: 5 Plots of the observed (a) histogram and estimated pdf’s and (b) estimated cdf’s for the
GBGMO and GGMOKw for example III.
In Tables 1, 2 and 3 the MLEs, se’s (in parentheses) and 95% confidence intervals (in parentheses)
of the parameters for the fitted distributions along with AIC, BIC, CAIC and HQIC values are
presented for example I, II, and III respectively. In the entire examples considered here based on the
lowest values of the AIC, BIC, CAIC and HQIC, the WBGMO distribution turns out to be a
better distribution than WGMOKw distribution. A visual comparison of the closeness of the fitted
densities with the observed histogram of the data for example I, II, and III are presented in the
figures 3, 4, and 5 respectively. These plots indicate that the proposed distributions provide a closer
fit to these data.
6. Conclusion
Beta Generalized Marshall-Olkin family of distributions is introduced and some of its important
properties are studied. The maximum likelihood and moment method for estimating the parameters
are also discussed. Application of three real life data fitting shows good result in favour of the
proposed family when compared to Generalized Marshall-Olkin Kumaraswamy extended family of
distributions. It is therefore expected that this family of distribution will be an important addition to
the existing literature on distribution.
Appendix: Maximum likelihood estimation for EBGMO
The pdf of the EBGMO distribution is given by
)(tf BGMOE
11
1
1
111
]1[][
),(1
n
t
tm
t
t
t
tt
ee
ee
eee
nmB
30
For a random sample of size n from this distribution, the log-likelihood function for the parameter
vector Tnm ),,,,( θ is given by
)(θ )],(log[)(log)1(logloglog00
nmBretrrrr
i
tr
ii
i
r
i
tt
r
i
ttr
i
t
ii
iii
een
eeme
0
00
]}1{[log)1(
]]}1{[1[log)1(]1[log)1(
The components of the score vector Tnm ),,,,( θ are
r
i
ttm
ii eenmrmrm
U0
]]}1{[1[log)()(
r
i
ttn
ii eenmrnrn
U0
]}1{[log)()(
U
r
i
tr
i
t ii eerr00
]1[log][loglog
r
itt
tttt
ii
iiii
eeeeeem
0 ]}1{[1}]1{[log]}1{[)1(
r
itt
tttt
ii
iiii
eeeeeen
0 ]}1{[]}1{[log]}1{[)1(
U
r
it
t
i
i
eer
0 1)1(
r
ittt
tt
iii
ii
eeeeem
0 ]1[]}{}1{[]1[][)1(
r
it
t
i
i
een
0 11)1(1
U
r
it
tr
it
tr
ii i
i
i
i
ee
eetr
000 1)1(
1)1(
r
ittt
t
iii
i
eeeem
0 ]1[]}{}1{[][)1(
r
itt
t
ii
i
eeen
0 ]1[)1(
The asymptotic variance covariance matrix for mles of the unknown parameters θ ),,,,( nm
of EBGMO ),,,,( nm distribution is estimated by
31
)ˆ(I 1n θ
)λ(var)α,λ(cov)θ,λ(cov)ˆ,λ(cov)ˆ,λ(cov
)λ,α(cov)α(var)θ,α(cov)ˆ,α(cov)ˆ,α(cov
)λ,θ(cov)α,θ(cov)θ(var)ˆ,θ(cov)ˆ,θ(cov
)λ,ˆ(cov)α,ˆ(cov)θ,ˆ(cov)ˆ(var)ˆ,ˆ(cov
)λ,ˆ(cov)α,ˆ(cov)θ,ˆ(cov)ˆ,ˆ(cov)ˆ(var
nm
nm
nm
nnnnmn
mmmnmm
Where the elements of the information matrix θθ
θθIˆ
2
n)()ˆ(ˆ
ji
l
can be derived using
the following second partial derivatives:
2
2
m )()( nmrmr
2
2
n )()( nmrnr
2
2
θ
2θr
r
itt
tttt
ii
iiii
eeeeeem
02θλλ
2λλθ2λλ
]]}α1{α[1[}]α1{α[log]}α1{α[)1(
r
itt
tttt
ii
iiii
eeeeeem
0θλλ
2λλθλλ
]}α1{α[1}]α1{α[log]}α1{α[)1(
2
2
α
r
it
t
i
i
eer
02λ
λ2
2 )α1()1θ(
αθ
r
i
ti
tti
ttt
beteeteeen
iiiiii
0
2λλ23λλ3λλ }α1{α2-}α1{α2-]α1[)1(θ
r
i
ti
tti
t
beteeten
iiii
0
λλ2λλ2 }α1{α}α1{α-)1(θ
r
i
ti
tti
ttt
beteeteeen
iiiiii
02
λλ2λλ2λλ }α1{α}α1{α-]α1[)1(θ
r
itt
tttttt
ii
iiiiii
ee
eeeeeem
0θλλ
2λλ23λλ31-θλλ
]}α1{α[1
}α1{2-}α1{α2]}α1{α[θ)1(
r
itt
tttttt
ii
iiiiii
eeeeeeeem
02θλλ
2λλ2λλ22-θ2λλ2
]]}α1{α[1[}α1{}α1{α]}α1{α[θ)1(
r
itt
tttttt
ii
iiiiii
eeeeeeeem
0θλλ
2λλ2λλ22-θλλ
]}α1{α[1}α1{}α1{α]}α1{α[1)-(θθ)1(
32
2
2
2λr
r
iti
t
ti
t
i
i
i
i
ete
ete
0λ
2λ
2λ
2λ22
α1λα
]α1[λα)1θ(
r
i
ti
tti
ti
beteetetn
iiii
0
λλ2λλ2 }α1{}α1{αα-α)1(θ
r
i
ti
tti
ti
tt
beteeteteen
iiiiii
0
λλ2λλ2λλ }α1{}α1{αα-]α1[)1(θ
r
i
ti
t
ti
tti
t
itt
bete
eteetetee
nii
iiii
ii
0
λ2λ
2λ2λ23λ2λ32λλ
}α1{α
}α1{αα3}α1{αα2-]α1[
)1(θ
r
itt
ttti
ttt
ii
iiiiii
eeeeeteeem
02θλλ
2λλ2λλ22-θ2λλ2
]]}α1{α[1[}α1{α}α1{αα-]}α1{α[θ)1(
r
itt
ttti
ttt
ii
iiiiii
eeeeeteeem
0θλλ
2λλ2λλ22-θλλ
]}α1{α[1}α1{α}α1{αα-]}α1{α[1)-(θθ)1(
r
itt
ti
t
ti
tti
ttt
ii
ii
iiii
ii
eeete
eteeteee
m0
θλλ
λ2λ
2λ2λ23λ2λ321-θλλ
]}α1{α[1}α1{α
}α1{αα3}α1{αα2-]}α1{α[θ
)1(
nm2 )( nmr
θ
2
m
r
itt
tttt
ii
iiii
eeeeee
0θλλ
λλθλλ
]}α1{α[1}]α1{α[log]}α1{α[
αm2
r
itt
tttttt
ii
iiiiii
eeeeeeee
0θλλ
λλ2λλ21-θλλ
]}α1{α[1]}α1{α[]}α1{α[]}α1{α[ θ
m2
r
itt
ti
tti
ttt
ii
iiiiii
eeeteeteee
0θλλ
λλ2λλ21-θλλ
]}α1{α[1]}α1{α[]}α1{αα[]}α1{α[ θ
θ
2
n ]}α1{α[log λλ
0
ii ti
tr
iete
α
2
n
r
i
tttttt
beeeeee iiiiii
0
λλ2λλ2λλ }α1{}α1{α-]α1[θ
33
λ
2
n
r
i
ti
tti
ttt
beteeteee iiiiii
0
λλ2λλ2λλ }α1{α}α1{αα-]α1[θ
αθ
2
αr
r
it
t
i
i
ee
0λ
λ
α1
r
i
tttttt
beeeeeen
iiiiii
0
λλ2λλ2λλ }α1{}α1{α-]α1[)1(
r
itt
tttttt
ii
iiiiii
eeeeeeeem
0θλλ
λλ2λλ21-θλλ
]}α1{α[1}α1{}α1{α-]}α1{α[)1(
r
itt
tttttttt
ii
iiiiiiii
eeeeeeeeee
02θλλ
λλλλ2λλ21-θ2λλ
]]}α1{α[1[}]α1{log[}α1{}α1{α-]}α1{α[m)-θ(1
r
itt
tttttttt
ii
iiiiiiii
eeeeeeeeee
0θλλ
λλλλ2λλ21-θλλ
]}α1{α[1}]α1{log[}α1{}α1{α-]}α1{α[m)-θ(1
λθ
2
r
iit
0]}α1{α[log λλ
0
ii ti
tr
iete
r
i
ti
tti
ttt
beteeteeen
iiiiii
0
λλ2λλ2λλ }α1{α}α1{αα-]α1[)1(
r
itt
ttti
ttt
ii
iiiiii
eeeeeteeem
0θλλ
λλ2λλ21-θλλ
]}α1{α[1}α1{}α1{αα-]}α1{α[)1(
r
itt
ttti
ti
tttt
ii
iiiiiiii
eeeeeteteeee
02θλλ
λλλλ2λλ21-θ2λλ
]]}α1{α[1[}]α1{log[}α1{}α1{αα-]}α1{α[m)-θ(1
r
itt
ttti
tti
ttt
ii
iiiiiiii
eeeeeteeteee
0θλλ
λλλλ2λλ21-θλλ
]}α1{α[1}]α1{log[}α1{}α1{αα-]}α1{α[m)-θ(1
2
r
it
it
ti
t
i
i
i
i
ete
ete
0λ
λ
2λ
λ2
α1]α1[α)1θ(
r
i
tttti
beeeetn
iiii
0
λλ2λλ2 }α1{}α1{α-)1(θ
r
i
tttti
tt
beeeeteen
iiiiii
0
λλ2λλ2λλ }α1{}α1{α-]α1[)1(θ
34
r
i
ti
tti
t
ti
tti
ttt
beteete
eteeteee
niiii
iiii
ii
0
λλ2λλ2
2λλ23λλ3λλ
}α1{}α1{α2
}α1{α}α1{αα2]α1[
)1(θ
r
itt
ti
tti
t
ti
tti
ttt
ii
iiii
iiii
ii
ee
eteete
eteeteee
m0 θλλ
λλ2λλ2
2λλ23λλ31θλλ
)}α1(α{1
}α1{}α1{α2
}α1{α}α1{αα2)}α1(α{θ
)1(
r
itt
ti
tti
t
tttttt
ii
iiii
iiiiii
ee
eteete
eeeeee
m0 2θλλ
λλ2λλ2
λλ2λλ22θ2λλ2
])}α1(α{1[
})α1{α}α1{αα(
})α1{}α1{α()}α1(α{θ
)1(
r
itt
ti
tti
t
tttttt
ii
iiii
iiiiii
ee
eteete
eeeeee
m0 θλλ
λλ2λλ2
λλ2λλ22θλλ
)}α1(α{1
})α1{α}α1{αα(
})α1{}α1{α()}α1(α{1)-θ(θ
)1(
Where (.) is the derivative of the digamma function.
References
1. Alizadeh M., Tahir M.H., Cordeiro G.M., Zubair M. and Hamedani G.G. (2015). The
Kumaraswamy Marshal-Olkin family of distributions. Journal of the Egyptian Mathematical
Society, in press.
2. Bornemann, Folkmar and Weisstein, Eric W. “Power series” From MathWorld-A Wolfram
Web Resource. http://mathworld.wolfram.com/PowerSeries.html.
3. Barreto-Souza W., Lemonte A.J. and Cordeiro G.M. (2013). General results for Marshall and
Olkin's family of distributions. An Acad Bras Cienc 85: 3-21.
4. Baharith L.A., Mousa S.A., Atallah M.A. and Elgyar S.H. (2014). The beta generalized
inverse Weibull distribution. British J Math Comput Sci 4: 252-270.
5. Cordeiro G.M., Silva G.O. and Ortega E.M.M. (2012a). The beta extended Weibull
distribution. J Probab Stat Sci 10: 15-40.
6. Cordeiro G.M., Cristino C.T., Hashimoto E.M. and Ortega E.M.M. (2013b). The beta
generalized Rayleigh distribution with applications to lifetime data. Stat Pap 54: 133-161.
7. Cordeiro G.M., Silva G.O., Pescim R.R. and Ortega E.M.M. (2014c). General properties for
the beta extended half-normal distribution. J Stat Comput Simul 84: 881-901.
8. Eugene N., Lee C. and Famoye F. (2002). Beta-normal distribution and its applications.
Commun Statist Theor Meth 31: 497-512.
35
9. George D. and George S. (2013). Marshall-Olkin Esscher transformed Laplace distribution
and processes. Braz J Probab Statist 27: 162-184.
10. Ghitany M.E., Al-Awadhi F.A. and Alkhalfan L.A. (2007). Marshall-Olkin extended Lomax
distribution and its application to censored data. Commun Stat Theory Method 36: 1855-
1866.
11. Ghitany M.E., Al-Hussaini E.K. and Al-Jarallah R.A. (2005). Marshall-Olkin extended
Weibull distribution and its application to censored data. J Appl Stat 32: 1025-1034.
12. Gieser P.W., Chang M.N., Rao P.V., Shuster J.J. and Pullen J. (1998). Modelling cure rates
using the Gompertz model with covariate information. Stat Med 17(8):831–839.
13. Greenwood J.A., Landwehr J.M., Matalas N.C. and Wallis J.R. (1979). Probability weighted
moments: definition and relation to parameters of several distributions expressable in inverse
form. Water Resour Res 15: 1049-1054.
14. Gui W. (2013a) . A Marshall-Olkin power log-normal distribution and its applications to
survival data. Int J Statist Probab 2: 63-72.
15. Gui W. (2013b). Marshall-Olkin extended log-logistic distribution and its application in
minification processes. Appl Math Sci 7: 3947-3961.
16. Gurvich M., DiBenedetto A. and Ranade S. (1997). A new statistical distribution for
characterizing the random strength of brittle materials. J. Mater. Sci. 32: 2559-2564.
17. Handique L. and Chakraborty S. (2015a). The Marshall-Olkin-Kumaraswamy-G family of
distributions. arXiv:1509.08108 [math.ST].
18. Handique L. and Chakraborty S. (2015b). The Generalized Marshall-Olkin-Kumaraswamy-G
family of distributions. arXiv:1510.08401 [math.ST].
19. Handique L. and Chakraborty S. (2016). Beta generated Kumaraswamy-G and other new
families of distributions. arXiv:1603.00634 [math.ST] , Under Review.
20. Handique L. and Chakraborty S. (2016b). The Kumaraswamy Generalized Marshall-Olkin
family of distributions.
21. Jones M.C. (2004). Families of distributions arising from the distributions of order statistics.
Test 13: 1-43.
22. Kenney J.F., Keeping and E.S. (1962). Mathematics of Statistics, Part 1. Van Nostrand, New
Jersey, 3rd edition.
23. Krishna E., Jose K.K. and Risti´c M. (2013). Applications of Marshal-Olkin Fréchet
distribution. Comm. Stat. Simulat. Comput. 42: 76–89.
24. Lehmann E.L. (1953). The power of rank tests. Ann Math Statist 24: 23-43.
25. Lemonte A.J. (2014). The beta log-logistic distribution. Braz J Probab Statist 28: 313-332.
36
26. Marshall A. and Olkin I. (1997). A new method for adding a parameter to a family of
distributions with applications to the exponential and Weibull families. Biometrika 84: 641-
652.
27. Mead M.E., Afify A.Z., Hamedani G.G. and Ghosh I. (2016). The Beta Exponential Frechet
Distribution with Applications, to appear in Austrian Journal of Statistics.
28. Moors J.J.A. (1988). A quantile alternative for kurtosis. The Statistician 37: 25–32.
29. Nadarajah S. (2005). Exponentiated Pareto distributions. Statistics 39: 255-260.
30. Nadarajah S., Cordeiro G.M. and Ortega E.M.M. (2015). The Zografos-Balakrishnan–G
family of distributions: mathematical properties and applications, Commun. Stat. Theory
Methods 44:186-215.
31. Nichols M.D. and Padgett W.J.A. (2006). A bootstrap control chart for Weibull percentiles.
Quality and Reliability Engineering International, v. 22: 141-151.
32. Pescim R.R., Cordeiro G.M., Demetrio C.G.B., Ortega E.M.M. and Nadarajah S. (2012). The
new class of Kummer beta generalized distributions. SORT 36: 153-180.
33. Sarhan A.M. and Apaloo J. (2013). Exponentiated modified Weibull extension distribution.
Reliab Eng Syst Safety 112: 137-144.
34. Singla N., Jain K. and Sharma S.K. (2012). The beta generalized Weibull distribution:
Properties and applications. Reliab Eng Syst Safety 102: 5-15.
35. Song K.S. (2001). Rényi information, loglikelihood and an intrinsic distribution measure.
EM J Stat Plan Infer 93: 51-69.
36. Weisstein, Eric W. “Incomplete Beta Function” From MathWorld-A Wolfram Web
Resource. http://mathworld.Wolfram.com/IncompleteBetaFunction.html.
37. Xu, K., Xie, M., Tang, L.C. and Ho, S.L. (2003). Application of neural networks in
forecasting engine systems reliability. Applied Soft Computing, 2(4): 255-268.
38. Zhang T. and Xie M. (2007). Failure data analysis with extended Weibull distribution.
Communication in Statistics – Simulation and Computation 36: 579-592.
![THE EXPONENTIATED GENERALIZED FLEXIBLE WEIBULL … · 2018. 9. 8. · Weibull family, Mudholkar and Srivastava [18], beta-Weibull distribution, Famoye et al. [6], generalized modified](https://img.pdfslide.us/doc/110x75/606a7b06ad36ab11840c32be/the-exponentiated-generalized-flexible-weibull-2018-9-8-weibull-family-mudholkar.jpg)
![The Matrix Generalized Inverse Gaussian …baner029/papers/16/CMCPMF.pdfMatrix Generalized Inverse Gaussian (MGIG) distributions [3,10] are a family of distributions over the space](https://img.pdfslide.us/doc/110x75/5f04904f7e708231d40e9764/the-matrix-generalized-inverse-gaussian-baner029papers16-matrix-generalized-inverse.jpg)
