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: subrata_stats@dibru.ac.in

(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, f (t ) and cdf F (t )  P [T  t ] , then its
Survival function (sf): F (t )  P [T  t ]  1  F (t ) ,
Hazard rate function (hrf): h(t )  f (t ) / F (t ) ,
Reverse hazard rate function (rhrf): r (t )  f (t ) / F (t ) ,

1

Cumulative hazard rate function (chrf): H (t )   log [ F (t )] ,


( p, q, r ) th Power Weighted Moment (PWM): p, q , r   t p [ F (t ) ] q [1  F (t ) ] r f (t ) dt ,




Rényi entropy: I R ( )  (1   ) 1 log  f (t )  dt  .
 

1.2 Beta-G family of distributions
The cdf of beta-G (Eugene et al. 2002 and Jones 2004) family of distribution is
F

BG

1
(t ) 
B (m, n)

F (t )

v

m 1

(1  v) n 1 d v 

o

B F ( t ) ( m, n )
B ( m, n )

(1)

 I F (t ) ( m , n )

t

Where I t ( m, n)  B ( m, n) 1  x m 1 (1  x) n 1 dx denotes the incomplete beta function ratio.
0

The pdf corresponding to (1) is
1
1
f BG (t ) 
f (t ) F (t ) m 1 [1  F (t )] n 1 
f (t ) F (t ) m 1 F (t ) n 1
B (m, n)
B ( m, n )
Where f (t )  d F (t ) dt .
B (m, n)  B F (t ) (m, n)
sf:
F BG (t ) 1  I F ( t ) (m, n ) 
B ( m, n )
hrf:

h BG (t )  f

BG

(t ) F BG (t ) 

rhrf:

r BG (t )  f BG (t ) F BG (t ) 

(2)

f (t ) F (t ) m 1 F (t ) n 1
B ( m, n )  B F ( t ) ( m, n )
f (t ) F (t ) m 1 F (t ) n 1
B F ( t ) ( m, n )

 B ( m, n )  B F ( t ) ( m, n ) 
H (t )   log 

B ( m, n )


Some of the well known beta generated families are the Beta-generated (beta-G) family (Eugene

chrf:

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 F GMO (t ) of
the GMOE family of distributions by exponentiation the sf of MOE family of distributions as

2



F

GMO

  G (t ) 
(t )  
 ,    t  ; 0    ; 0    
1   G ( t ) 

(3)

where    t  ,   0 (   1   ) and   0 is an additional shape parameter. When   1,
F GMO (t )  F MO (t ) and for     1,

F GMO (t )  F (t ) . The cdf and pdf of the GMOE distribution

are respectively
F

and

f

GMO

GMO

  G (t ) 
(t )   

1   G (t ) 

  G (t ) 
(t )  1  

1   G (t ) 

 1



(4)

  g (t )     g (t ) G (t )  1


2
[1   G (t )]  1
[1   G (t )] 

(5)

Reliability measures like the hrf, rhrf and chrf associated with (1) are
h

GMO

f GMO (t )    g (t ) G (t )  1
(t )  GMO

F
(t )
[1   G (t )]  1



r

GMO



  G (t ) 
1   G (t ) 





g (t )
1
h(t )

G (t ) 1   G (t )
1   G (t )

f GMO (t )    g (t ) G (t )  1
(t )  GMO 
F
(t )
[1   G (t )]  1

  G (t ) 
1 

1   G (t ) 



   g (t ) G (t )  1
   g (t ) G (t )  1

[1   G (t )] [{1   G (t )}     G (t )  ] [1   G (t )]  1    G (t )  [1   G (t )]


H

GMO

  G (t ) 
  G (t ) 
  log 
(t )   log 


1   G (t ) 
1   G (t ) 

Where g (t ) , G (t ) , G (t ) and h (t ) are respectively the pdf, cdf, sf and hrf of the baseline distribution.
We denote the family of distribution with pdf (3) as GMOE( , , a, b ) which for   1, reduces
to MOE( , a, b) .
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), MarshallOlkin 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 MarshallOlkin family of distribution (Handique and Chakraborty 2016b).

3

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 ( BGMO  G ) 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 f (t ) and F (t ) respectively in the Beta
formulation in (2) and call it BGMO  G distribution. The pdf of BGMO  G is given by

f

BGMOG

1    g (t ) G (t )  1    G (t ) 
(t ) 
1 
B (m, n) [1   G (t )]  1  1   G (t ) 








m 1

   G (t )   

 
 1   G (t )  

n 1

(6)

, 0  t  , 0  a, b  , m  0, n  0
Similarly substituting from equation (4) in (1) we get the cdf of BGMO  G respectively as
F BGMOG (t )  I

  G (t ) 
1 

 1  G ( t ) 



(m, n)

(7)

The sf, hrf, rhrf and chrf of BGMO  G distribution are respectively given by

F BGMOG (t )  1  I

sf:

  G (t ) 
1 

 1 G ( t ) 



(m, n)

hrf :
h BGMOG (t ) 

1
   g (t ) G (t )  1
B ( m, n) [1   G (t )]  1 [1  I 

 G (t ) 
1 

 1  G ( t ) 

   G (t )   
 1  
 
 1   G (t )  

m 1

   G (t )   

 
 1   G (t )  

rhrf :

4



( m, n ) ]

n 1

(8)

r BGMOG (t ) 

1
   g (t ) G (t )  1
B (m, n) [1   G (t )]  1 [ I
 ( m, n) ]
  G (t ) 
1 

 1  G ( t ) 

   G (t )   
 1  
 
 1   G (t )  
chrf:

H BGMOG (t )   log [ 1  I

m 1

   G (t )   

 
 1   G (t )  

  G (t ) 
1 

 1 G ( t ) 



n 1

(9)

(m, n) ]

Remark: The BGMO G (m, n,  , ) reduces to
(i)

BMO (m, n, ) (Alizadeh et al., 2015) for   1 ;
Mathew, 2008)

(ii) GMO ( , ) ,(Jayakumar and

if m  n  1 ; (iii) MO ( ) (Marshall and Olkin, 1997)

when

m  n    1 ; and (iv) B (m, n) (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 T1 , T 2 ...,T m  n 1 be a sequence of i.i.d. random variables with cdf 1  [1  G (t ) a ] b . Then
the pdf of the mth order statistics T ( m ) is given by


( m  n  1) !
[1  { G (t ) 1   G (t ) }  ] m 1 [{ G (t ) 1   G (t ) }  ] ( m  n 1)  m
(m  1)! [( m  n  1)  m] !
    g (t ) G (t )  1 [1   G (t )]  1



 ( m  n)
[1  { G (t ) 1   G (t ) }  ] m 1 [{ G (t ) 1   G (t )}  ] n 1
 ( m)  ( n )
    g (t ) G (t )  1 [1   G (t )]  1



1
[1  { G (t ) 1   G (t ) }  ] m 1 [{ G (t ) 1   G (t ) }  ] n 1
B ( m, n )
    g (t ) G (t )  1 [1   G (t )]  1

2.2 Shape of the density function
Here we have plotted the pdf of the BGMO  G for some choices of the distribution G and parameter
values to study the variety of shapes assumed by the family.

5

2.5
1.5

2.0

m=1,n=2,=1.5,=2.1,=0.9,=0.6
m=13,n=3.5,=2.5,=0.6,=0.5,=0.5
m=50,n=8.5,=2.2,=0.89,=0.5,=0.5
m=0.05,n=0.45,=1.4,=1.5,=0.5,=0.5
m=0.2,n=0.4,=3.5,=1.5,=0.5,=0.4
m=40,n=9.5,=2.5,=0.5,=0.5,=0.5

0.0

0.0

0.5

1.0

Density

1.5

2.0

m=0.6,n=2,=2,=0.5,=0.9
m=1.3,n=1.9,=1.6,=0.6,=0.1
m=3.5,n=2,=0.5,=1.6,=0.5
m=19.5,n=1.9,=0.9,=1.6,=0.6
m=19.5,n=1.9,=0.9,=1.6,=0.5
m=2.9,n=2.2,=2.5,=0.5,=0.5

0.5

Density

BGMO-W

1.0

2.5

BGMO-E

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

X

(a)

(b)

3.0

3.5

2.5
2.0
1.5

Density

1.5

0.5
0.0

0.0

0.5

1.0

m=0.1,n=2.5,=1.4,=3.5,=0.5,=0.5
m=0.3,n=2.1,=1.3,=4.5,=0.5,=0.5
m=0.23,n=4.45,=1.3,=4.5,=0.5,=0.5
m=0.26,n=3.35,=1.4,=4.5,=0.3,=0.5
m=0.05,n=2.5,=1.2,=4.5,=0.5,=0.5
m=0.2,n=2.5,=1.3,=4.5,=0.5,=0.5

1.0

2.5

2.5

BGMO-Fr

m=8.2,n=1.9,=5.55,=3.5,=0.48,=0.8
m=7.2,n=1.9,=5.45,=3.1,=0.42,=0.7
m=5,n=1.56,=2.5,=0.5,=0.5,=0.5
m=5.2,n=1.56,=4.5,=2.5,=0.5,=0.5
m=8.2,n=1.9,=5.55,=3.2,=0.45,=0.6
m=1,n=3.5,=2.5,=0.5,=0.5,=0.5

2.0

2.0

X

BGMO-L

Density

1.5

0

1

2

3

4

0

1

2

3

4

X

X

(c)

(d)

5

6

Fig 1: Density plots (a) BGMO  E , (b) BGMO  W , (c) BGMO  L and (d) BGMO  Fr
Distributions:

6

7

BGMO-W
2.5

3.0

BGMO-E

2.0
h(x)

m=0.01,n=0.65,=1, =1.9,=0.9,=0.8
m=0.08,n=0.3,=2.1, =1.83,=0.8,=0.7
m=0.04,n=0.25,=1.7,=2,=0.8,=1.8
m=0.05,n=0.45,=1.5,=1.8,=0.8,=0.8
m=0.08,n=0.15,=2, =2.2,=0.8,=1.8
m=0.02,n=0.65,=0.9,=1.9,=0.9,=0.8

0.0

0.0

0.5

0.5

1.0

1.0

1.5

h(x)

1.5

2.0

2.5

m=0.05,n=1.9,=1.6,=0.9,=0.7
m=23.5,n=1.99,=0.58,=1.6,=0.5
m=0.04,n=1.99,=0.58,=1.6,=0.5
m=19.5,n=1.84,=0.9,=2.8,=0.9
m=15.5,n=1.9,=0.8,=1.6,=0.5
m=0.89,n=2.78,=0.48,=1.5,=0.5

1.0

1.5

2.0

2.5

0

1

2

3

X

X

(a)

(b)

BGMO-L

BGMO-Fr
2.5

0.5

2.5

0.0

1.5

2.0

m=0.6,n=1.9,=0.25,=5.5,=0.8,=0.9
m=0.99,n=2.5,=3,=3.5,=0.5,=0.5
m=5.23,n=4.65,=2.3,=6.7,=0.5,=0.5
m=1.9,n=2.5,=3,=3.5,=0.63,=0.7
m=0.3,n=1.8,=0.2,=3.5,=0.5,=0.8
m=7.23,n=4.95,=2.3,=6.7,=0.41,=0.5

0.0

0.5

1.0

h(x)
0.0

0.5

1.0

h(x)

1.5

2.0

m=2,n=0.35, =2.5,=5.5,=2.9,=1.9
m=7.2,n=1.9,=5.45,=3.1,=0.42,=0.7
m=5,n=1.56, =2.5,=0.5,=0.5,=0.5
m=4,n=1.56, =4.5,=2.5,=0.5,=0.5
m=4.2,n=1.99,=6.55,=9.2,=0.45,=0.6
m=0.5,n=3.8,=0.19,=3.5,=0.6,=0.9

4

0

1

2

3

4

0

X

1

2

3

4

5

6

7

X

(c)
(d)
Fig 2: Hazard plots plots (a) BGMO  E , (b) BGMO  W , (c) BGMO  L and (d) BGMO  Fr
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 BGMO  G distributions
Some special cases of the BGMO  G family of distributions are presented in this section.
3.1 The BGMO  exponential ( BGMO  E ) distribution
 t
, t  0
Let the base line distribution be exponential with parameter   0, g ( t :  )   e

and G (t :  )  1  e  t ,

t  0 then for the BGMO  E model we get the pdf and cdf respectively as:

7

1     e  t [ e  t ]  1    e  t 
f BGMOE (t ) 
1 
B (m, n) [1   e t ]  1  1   e t 


F BGMOE (t )  I

and

F BGMOE (t )  1  I

sf:

  e  t 
1 

t
 1  e 

  e t
1 
t
 1  e















m 1

   e  t   


 t 
 1   e  

n 1

( m, n )

(m, n)

hrf :

h BGMOE (t ) 

1
    e  t [ e t ]  1
B (m, n) [1   e  t ]  1 [1  I  t   (m, n ) ]
e
1 

 t
1  e 

   e  t   
 1  

 t 
 1   e  

m 1

   e  t   


 t 
 1   e  

n 1

rhrf :

r BGMOE (t ) 

1
    e t [ e  t ]  1
B (m, n) [1   e  t ]  1 [ I  t   (m, n) ]
e
1 

t
 1 e 

   e  t   
 1  

 t 
 1   e  
H BGMOE (t )   log [ 1  I

chrf:

m 1

   e  t   


 t 
 1   e  

  e t
1 
 t
1  e







n 1

( m, n ) ]

3.2 The BGMO  Lomax ( BGMO  L ) distribution
Considering the Lomax distribution (Ghitany et al. 2007) with pdf and cdf given
by g (t :  , )  (   ) [1  (t  )]  (  1) , t  0 , and G ( t :  , )  1  [1  ( t  ) ]   ,   0 and   0 the
pdf and cdf of the BGMO  L distribution are given by

f BGMOL (t ) 

1    (   )[1  (t  )]  (  1) [ [1  ( t  ) ]   ]  1
B (m, n)
[1   [1  ( t  ) ]   ]  1

   [1  ( t  ) ]     
 1  

 
 1   [1  ( t  ) ]  
and

sf:

F BGMOL (t )  I

m 1

   [1  ( t  ) ]     


 
 1   [1  ( t  ) ]  

  [ 1 ( t  ) ]   
1 


 1 [ 1 ( t  ) ] 

F BGMOL (t )  1  I



( m, n )

  [ 1  ( t  )]   
1 
 

 1   [ 1  ( t  )]

8



(m , n )

n 1

hrf : h BGMOL (t )



1    (   ) [1  (t  )]  (  1) [1  (t  )]   ( 1)
B ( m, n )
[1   {1  (t  )}   ]  1
1 I

1

  [ 1 ( t  )]   
1 


 1 [ 1 ( t  )] 

   [1  (t  )]     
 1  

 
 1   [1  (t  )]  
rhrf:



m 1

   [1  (t  )]     


 
 1   [1  (t  )]  

( m, n )

n 1

r BGMOL (t )

1    (   )[1  (t  )]  (  1) [1  (t  )]   ( 1)
B ( m, n )
[1   {1  (t  )}   ]  1
I
   [1  (t  )]     
 1  

 
 1   [1  (t  )]  

chrf:



H BGMOL (t )   log [ 1  I

m 1

1

   [1  (t  )]     


 
 1   [1  (t  )]  

  [ 1 ( t  )]   
1 


1  [ 1 ( t  )] 





  [ 1 ( t  )]   
1 


 1 [ 1 ( t  )] 

(m, n)

n 1

( m, n ) ]

3.3 The BGMO  Weibull ( BGMO  W ) distribution
Considering the Weibull distribution (Ghitany et al. 2005, Zhang and Xie 2007) with parameters




  0 and   0 having pdf and cdf g (t )    t  1 e   t and G (t )  1  e   t respectively we get
the pdf and cdf of BGMO  W distribution as


f BGMOW (t ) 

 1

 t 

1   t e
[e
  t   1
B ( m, n )
[1   e
]

and

F BGMOW (t )  I

sf:

F BGMOW (t )  1  I

hrf:

 t 

  e  t 
1 

 t 
 1  e








  e  t 
1 

 t 
 1  e

]

   e  t 
1  

 1   e   t


 1





( m, n )








( m, n )

h BGMOW (t )




1      t  1 e   t [ e   t ]  1


B (m, n)
1 I
[1   e   t ]  1

   e  t    

 1  
 
 1   e   t  



m 1

1
  e  t 
1 

 t 
 1  e








( m, n )

  e  t    


 
 1   e   t  



9

n 1








m 1

  e  t 


 1   e   t













n 1

rhrf:


r

BGMOW



1
     t  1 e  t [ e  t ]  1
(t ) 

B (m, n)
I
[1   e  t ]  1

1

  e  t 
1 
1 e   t 


   e  t    

 1  
 
 1   e  t  



H BGMOW (t )   log [ 1  I

chrf:

  e  t 
1 

 t 
 1  e

m 1








   e  t    


 
 1   e  t  










(m, n)

n 1

(m, n) ]

3.4 The BGMO  Frechet ( BGMO  Fr ) distribution
Suppose the base line distribution is the Frechet distribution (Krishna et al., 2013) with pdf and cdf
g (t )     t  (  1) e  ( t )

given by





G (t )  e  ( t ) , t  0

and

respectively, and then the

corresponding pdf and cdf of BGMO  Fr distribution becomes


f

BGMOFr

1       t  (  1) e  ( t ) [1  e  (
(t ) 

B ( m, n )
[1   [1  e  ( t ) ] ]  1
   [1  e  (  t )  ]   
 1  
 

 1   [1  e  ( t ) ]  



F BGMOFr (t )  I

and

F BGMOFr (t )  1  I

sf:

hrf :

m 1

]  1

   [1  e  ( t )  ]


 1   [1  e  ( t )


  [ 1 e (  t )  ] 

1 

 ( t )  
]
 1  [ 1 e

  [ 1 e  (  t )  ] 

1 

(  t )  
]
 1  [ 1 e

t) 







] 








n 1

(m, n)

( m, n )

h BGMOFr (t ) 




1       t  (  1) e  ( t ) [1  e  ( t ) ]  1

B ( m, n )
1 I
[1   [1  e  ( t ) ] ]  1

1

  [ 1 e (  t )  ] 

1 

(  t )  
]
 1 [ 1 e

   [1  e  (  t )  ]   
 1  
 

 1   [1  e  ( t ) ]  


rhrf :

r BGMOFr (t ) 

10

m 1



   [1  e  (  t )  ]


 1   [1  e  ( t )


( m, n )



] 








n 1





1
      t  ( 1) e  ( t ) [1  e ( t ) ]  1

B ( m, n )
I
[1   [1  e ( t ) ] ]  1
   [ 1  e ( t )  ]   
 1  
 

 1   [1  e ( t ) ]  


chrf:

H BGMOFr (t )   log [ 1  I

  [ 1 e  (  t )  ] 

1 

(  t )  
]
 1  [ 1 e



1
  [1 e (  t )  ] 

1 
 1 [1 e (  t )  ] 



m 1



( m, n )

   [ 1  e ( t )  ]


 1   [1  e ( t )




] 








n 1

( m, n ) ]

3.5 The BGMO  Gompertz ( BGMO  Go ) distribution
Next by taking the Gompertz distribution (Gieser et al. 1998) with pdf and cdf g (t )   e t e
and

G (t )  1  e



 t
( e 1)


,   0,   0, t  0

respectively,

we

get

the

pdf

and

BGMO  Go distribution as
f BGMOGo (t ) 


t



 t
( e 1)




 t
( e 1)
 1


1   e e
[e
 t

(
e
1)
B (m, n)
[1   e 
]  1

]

and

F BGMOGo (t )  I

sf:

F BGMOGo (t )  1  I

hrf:



 
 ( e t 1)  
  e 
 
1  

 t
 ( e 1) 
 1   e 

 
 



 ( et 1)
 e 
1 

 ( et 1 )


 1  e




 ( e t 1)  
  e 
 


 t
 ( e 1) 
 1   e 

 


( m, n )





 ( et 1)
 e 
1 

 ( et 1 )


1  e









( m, n )

h BGMOGo (t ) 

 ( et 1)


 ( et 1)

1
    e t e 
[e 
 t
 ( e 1)
B ( m, n )
[1   e 
]  1

]  1



 
 ( et 1)
 


e
 
 1  

 t

 ( e 1) 
 1   e 

 
 

rhrf:







m 1

1
1 I

m1



 ( e t 1 ) 
 e 

1 


 ( e t 1 ) 


 1 e




(m, n)




 ( et 1)  

  e
 


 t
 ( e 1) 
 1   e 

 


r BGMOGo (t ) 

11

n 1

n 1



 t
( e 1)


cdf

of



t



 t
( e 1)




 t
( e 1)
 1


1   e e
[e

 ( e t 1)
B (m, n)
[1   e 
]  1

]



 
 ( e t 1)  
  e 
 
 1 



t

 ( e 1) 
 1   e 

 
 

chrf:

1
I

H BGMOGo (t )   log [ 1  I

m 1



 ( e t 1)
 e 
1 

 ( et 1 )


 1  e









( m, n )




 ( e t 1)
 
  e 
 


 t
 ( e 1) 
 1   e 

 




 ( et 1 )
 e 
1 

 ( e t 1)


 1 e









n 1

( m, n ) ]

3.6 The BGMO  Extended Weibull ( BGMO  EW ) distribution
The pdf and the cdf of the extended Weibull (EW) distributions of Gurvich et al. (1997) is given by

t  D  R ,   0

g (t :  , )   exp[ Z (t :  )] z (t :  ) and G (t :  , )  1  exp[ Z (t :  )] ,

where Z (t :  ) is a non-negative monotonically increasing function which depends on the parameter
vector  . and z (t :  ) is the derivative of Z (t :  ) .
By considering EW as the base line distribution we derive pdf and cdf of the BGMO  EW as

f BGMOEW (t ) 

1     exp[ Z (t :  )] z (t :  )[ exp{ Z (t :  )} ]  1
B ( m, n )
[1   exp{   Z (t :  )}]  1

   exp[ Z (t :  )]   
 1  
 
 1   exp[ Z (t :  )]  
and

F BGMOEW (t )  I

m 1

  exp[   Z ( t : )] 
1 

 1  exp[   Z ( t : )] 

   exp[ Z (t :  )]   

 
 1   exp[ Z (t :  )]  



n 1

( m, n )

Important models can be seen as particular cases with different choices of Z (t :  ) :
(i) Z (t :  ) = t : exponential distribution.
(ii) Z (t :  ) = t 2 : Rayleigh (Burr type-X) distribution.
(iii) Z (t :  ) = log (t k ) : Pareto distribution
(iv) Z (t :  ) =  1[exp (  t )  1] : Gompertz distribution.
sf:

F BGMOEW (t )  1  I

  exp[   Z ( t : )] 
1 

1  exp[   Z ( t : )] 



( m, n )

12

hrf: h BGMOEW (t ) 

1     exp[ Z (t :  )] z (t :  )[ exp{ Z (t :  )}]  1
B (m, n)
[1   exp{   Z (t :  )}]  1
1 I

1

  exp[  Z ( t : )] 
1 

 1 exp[   Z ( t : )] 

   exp[ Z (t :  )]   
 1  
 
 1   exp[ Z (t :  )]  

m 1



( m, n )

   exp[ Z (t :  )]   

 
 1   exp[ Z (t :  )]  

n 1

rhrf: r BGMOEW (t ) 

1     exp[ Z (t :  )] z (t :  )[ exp{ Z (t :  )} ]  1
B (m, n)
[1   exp{   Z (t :  ) }]  1
1 I
   exp[ Z (t :  )]   
 1  
 
 1   exp[ Z (t :  )]  
chrf:

H BGMOEW (t )   log [ 1  I

m 1

  exp[   Z ( t : )] 
1 

 1 exp[  Z ( t : )] 



1
  exp[   Z ( t : )] 
1 

 1  exp[   Z ( t : )] 



( m, n )

  exp[ Z (t :  )]   

 
1   exp[ Z (t :  )]  

n 1

( m, n ) ]

3.7 The BGMO  Extended Modified Weibull ( BGMO  EMW ) distribution
The modified Weibull (MW) distribution (Sarhan and Zaindin 2013) with cdf and pdf is given by

G (t ; ,  ,  )  1  exp [   t   t ] , t  0,   0, ,   0,    0 and

g (t ; ,  ,  )  (    t  1 ) exp[   t   t  ] respectively.
The corresponding pdf and cdf of BGMO  EMW are given by
f BGMOEMW (t ) 

1    (    t  1 ) exp [   t   t  ] [ exp{   t   t  } ]  1
B ( m, n )
[1   exp {   t   t  }]]  1

   exp [   t   t  ]   
 1  

 
 1   exp [   t   t ]  
F BGMOEMW (t )  I

and

sf:

F BGMOEMW (t )  1  I

m 1

   exp [   t   t  ]   


 
 1   exp [   t   t ]  

  exp [  t   t  ] 
1 
 
 1  exp [   t   t ] 

  exp [  t   t  ] 
1 
 
 1  exp [   t   t ] 

hrf : h BGMOEMW (t ) 

13





(m, n)

(m, n)

n 1

1    (    t  1 ) exp [   t   t  ] [ exp {   t   t  } ]  1
B (m, n)
[1   exp {   t   t  }]]  1
1 I
   exp [   t   t  ]   
 1  

 
 1   exp [   t   t ]  

m 1

1
  exp [   t   t  ] 
1 
 
 1  exp [  t   t ] 

   exp [   t   t  ]   


 
 1   exp [   t   t ]  



( m, n )

n 1

rhrf: r BGMOEMW (t ) 
1
   (     t  1 ) exp [   t   t  ] [ exp {   t   t  } ]  1
B ( m, n)
[1   exp {   t   t  }]]  1
I
   exp [   t   t  ]
 1  

 1   exp [   t   t

chrf:

H BGMOEMW (t )   log [ 1  I



]







m 1

1
  exp[  t   t  ] 
1 
 
 1 exp [  t   t ] 

  exp [   t   t  ]


1   exp [   t   t

  exp [   t   t  ] 
1 
 
 1  exp [  t   t ] 





]









( m, n)

n 1

( m, n ) ]

3.8 The BGMO  Extended Exponentiated Pareto ( BGMO  EEP ) distribution
The pdf and cdf of the exponentiated Pareto distribution, of Nadarajah (2005), are given respectively
k 
by g (t )   k  k t  ( k 1) [1  ( t ) k ]  1 and G ( t )  [1  ( t ) ] , x   and  , k ,   0 . Thus the

pdf and the cdf of BGMO  EEP distribution are given by

f

BGMOEEP

1     k  k t  ( k 1) [1  ( t ) k ]  1 [1  {1  ( t ) k }  ]  1
(t ) 
B (m, n)
[1   [1  {1  ( t )k }  ]]  1
   [1  {1  ( t )k }  ]   
 1  

k  
 1   [1  {1  ( t ) } ]  

and

sf:

F BGMOEEP (t )  I
F BGMOEEP (t )  1  I

m 1

   [1  {1  ( t )k }  ]   


k  
 1   [1  {1  ( t ) } ]  

  [ 1{1 ( t ) k }  ] 
1 

k 
 1  [ 1{1 ( t ) } ] 

  [ 1{1 ( t ) k }  ] 
1 

k 
 1  [ 1{ 1 ( t ) } ] 





n 1

(m, n)

( m, n )

hrf: h BGMOEEP (t ) 

1     k  k t  ( k 1) [1  ( t ) k ]  1 [1  {1  ( t )k }  ]  1
B ( m, n )
[1   [1  {1  ( t ) k }  ]]  1
1 I

1

  [ 1{1 ( t ) k }  ] 
1 

k 
 1 [ 1{ 1 ( t ) } ] 

   [1  {1  ( t )k }  ]   
 1  

k  
 1   [1  {1  ( t ) } ]  

m 1

   [1  {1  ( t ) k }  ]   


k  
 1   [1  {1  ( t ) } ]  

14

n 1



( m, n )

rhrf: r BGMOEEP (t ) 

1
    k  k t  ( k 1) [1  ( t ) k ]  1 [1  {1  ( t ) k }  ] 1
B (m, n)
[1   [1  {1  ( t ) k }  ]]  1
I
   [1  {1  ( t ) k }  ]
 1  
k 
 1   [1  {1  ( t ) }
chrf:



]







m 1

   [1  {1  ( t ) k }  ]

k 
 1   [1  {1  ( t ) }

H BGMOEEP (t )   log [ 1  I

  [ 1{1 ( t ) k }  ] 
1 

k 
 1  [ 1{1 ( t ) } ] 



1
  [1{1 ( t ) k }  ] 
1 
k  
 1 [1{1 ( t ) } ] 



]









( m, n )

n 1

( m, n ) ]

4. General results for the Beta Generalized Marshall-Olkin ( BGMO  G ) family of
distributions
In this section we derive some general results for the proposed BGMO  G family.
4.1 Expansions
By using binomial expansion in (6), we obtain
f BGMOG (t; , , m, n)

1    g (t ) G (t )  1    G (t ) 

1 
B (m, n) [1   G (t )]  1  1   G (t ) 



 g (t )   G (t )


B (m, n) [1   G (t )] 2  [1   G (t )]

 1











m 1

   G (t )   

 
 1   G (t )  

   G (t )   
1  
 
 1   G (t )  

m 1

n 1

   G (t )   

 
 1   G (t )  




f MO (t , ) [ F MO (t , )]  1 [1  {F MO (t , )}  ] m1 [ F MO (t ,  ) ]  ( n1)
B ( m, n )




f MO (t , )[ F MO (t , )]  n 1 [1  {F MO (t , )}  ] ( m 1)
B ( m, n )



m 1 m  1



 ( 1) j [ F MO (t; )]  j
f MO (t , ) [ F MO (t , )]  n 1  
j 
B ( m, n )
j 0 



m 1 m  1



 (1) j [ F MO (t ; )]  ( jn )1
f MO (t ,  )  
j 
B ( m, n )
j 0 
m 1

 f MO (t , )   j [ F MO (t ; )]  ( j  n ) 1

(10)

j 0

m 1



j

d

  ( j  n) dt [ F

MO

(t ; )]  ( j  n )

j 0

15

n 1

m 1

d
[ F MO (t; )]  ( j  n)
dt

   j
j 0

m 1



d

   dt [ F

MO

j

(11)

(t ; ( ( j  n )))]

j 0

m 1



f MO (t; ( ( j  n )))

 

j

(12)

j 0

Where  j 

 m  1
(1) j 1

 and  j   j  ( j  n)
B (m, n) ( j  n)  j 

Alternatively, we can expand the pdf as
m 1

 f MO (t , )   j [ F MO (t; )]  ( j  n ) 1
j 0

 ( j  n ) 1

m 1

 f MO (t , )  



j

j 0

l 0

  ( j  n )  1

 (1) l [ F MO (t ; ) ] l
l



 ( j  n ) 1

 f MO (t , )



l

[ F MO (t ; ) ] l

(13)

l 0

 ( j  n ) 1




l 0

l d
[ F MO (t; ) ] l 1
l  1 dt

 ( j  n )1



d

   dt [ F
l

MO

(t ; ) ] l 1

l 0


  ( j  n)  1
  ( j  n )  1
1 
Where  l    j (1) l 
 and  l 

 j (1) l 

l
l
l  1 j 0
j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)
(1  b) n



B ( z ; a, b )  B z ( a , b)  z a 
n 0

n ! ( a  n)



 za 
n 0

zn

Where (x) n is a Pochhammer symbol.

(1) n (b  1) !
zn
n ! (b  n  1) ! (a  n)


 b  1 (1) n n

 z a  
z
n 0  n  ( a  n)

F BGMOG (t )  I

  G (t ) 
1 

 1 G ( t ) 



(14)

(m, n)

Using (14) in (7) we have

16

   G (t )   
1
 

1  
B (m, n)   1   G (t )  






i 0



i 0

i

 n  1
(1) i

 [1  { F MO (t , ) }  ] mi
B (m, n) (m  i )  i 
 n  1   m  i 
(1) i

 
 (1) j [ F MO (t , ) ]  j
B (m, n ) (m  i)  i  j  0  j 





i, j 0

 n  1  m  i  j
(1) i  j



B (m, n) (m  i)  i   j  k  0






 n  1 (1) i    G (t )  
 


1  

i  (m  i)   1   G (t )  
i 0 





 n  1 (1) i
1

[1  { F MO (t , ) }  ] m  
[1  { F MO (t , ) }  ] i
i  (m  i )
B ( m, n )
i 0 





m

j



i, j 0 k 0



k

 n  1  m  i  
(1) i  j  k



B (m, n) (m  i )  i   j   k

j
 (1) k [ F MO (t , ) ] k

j  MO
 [ F (t ,  ) ] k


By exchanging the indices j and k in the sum symbol, we have


F BGMO (t ; , , m, n) 





i, k  0 j k

(1) i  j  k  n  1  m  i   



B (m, n) (m  i )  i   j   k

j  MO
 [ F (t , ) ] k


and then


F BGMO (t ; , , m, n) 



k

F MO (t , ) k

(15)

k 0



 n  1  m  i  
(1) i  j  k



i   j   k
j  k B ( m, n ) ( m  i ) 


Where  k   
i 0

j



Similarly an expansion for the cdf of BGMO  G can be derives as

F BGMOG (t ; , , m, n)  I1[ F MO (t , ) ]  (m, n)
m  n 1





pm
m  n 1





pm
m  n 1



 m  n  1

 [1  { F MO (t , )}  ] p [ F MO (t , )  ] m  n 1 p
p


 m  n  1 p  p 

    (1) q {F MO (t , )}  q [ F MO (t , )  ] m  n 1 p
p

 q 0  q 
p

  (1)
p m q 0
m  n 1





pm
m  n 1



q

 p   m  n  1 MO
  
 [ F (t , )]  ( m  n 1 p  q )
q
p
 


 ( m  n 1 p  q )
 ( m  n  1  p  q ) 
q  p   m  n  1





 (1) r [ F MO (t , )] r
(

1
)


 q 

p 
r
q 0
r 0
 


p

p  ( m  n 1 p  q )

 
p m q 0


r0

 p   m  n  1  (m  n  1  p  q )  MO
 
 [ F (t , )] r
(1) q  r   
p 
r
 q 

17

m  n 1



p  ( m  n 1 p  q )

 



p m q 0

Where 



p,q,r

[ F MO (t , )] r

(16)

r 0

p, q , r

 p   m  n  1   ( m  n  1  p  q ) 


 (1) q  r   
p  
r
 q 


4.2 Order statistics
Suppose T1, T 2 ,...T n is a random sample from any BGMO  G distribution. Let T r : n denote the r th
order statistics. The pdf of T r : n can be expressed as
n!
f
(r  1)! (n  r )!

f r : n (t ) 



BGMOG

(t ) F BGMOG (t ) r 1 {1  F BGMOG (t )} n  r

nr
n!
(1) j

(r  1)! (n  r )! j  0

n  r 

 f
 j 

BGMOG

(t ) F BGMOG (t ) j  r 1

Now using the general expansion of the BGMO  G distribution pdf and cdf we get the pdf of the

r th order statistics for of the BGMO  G is given by
f r : n (t ) 

nr
n!
 (1) j
(r  1)! (n  r )! j  0

 n  r   MO

  f (t , )
 j 

 ( j  n ) 1


l 0

l


[ F MO (t; ) ] l 


 
MO
k
   k F (t ,  ) 
 k 0


j  r 1

Where  l and  k defined in above
 

Now    k F MO (t , ) k 
 k 0


Where

d j  r 1, k 

j  r 1



  d j  r 1, k [ F MO (t , ) ] k
k0

1 k
 [c ( j  r )  k ]  c d j  r 1,
k  0 c 1

(Nadarajah et. al 2015)

k c

Therefore the density function of the r th order statistics of BGMO  G distribution can be
expressed as
f r : n (t ) 

nr
n!
 (1) j
( r  1)! ( n  r ) ! j  0

 n  r   MO

  f (t , )
 j 

 ( j  n ) 1


l 0

l


[ F MO (t; ) ] l 






  d j  r 1, k [ F MO (t , ) ] k 
 k 0



nr
n!
( 1) j

(r  1) ! ( n  r )! j  0

 ( j  n ) 1 

 n  r   MO

  f (t , )    l d j  r 1, k [ F MO (t , ) ] k  l 
l 0
k 0
 j 


 ( j  n )1 

 f MO (t ,  )

 
l 0

l,k

[ F MO (t , ) ] k l

(17)

k 0

18

Where  l , k 

n r
n!
(1) j

(r  1)! (n  r )! j  0

n  r

  l d j  r 1, k and  l and d j  r 1, k defined above.
 j 

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 ( p, q, r )th PWM of T
is defined by


p , q , r   t p F (t ) q [1  F (t )] r f (t ) dt


From equations (10) and (13) the s th moment of T can be written either as


E (T s ) 

t

s

f BGMOG (t ; , , m, n ) dt





m1

 

j

t

j 0



m 1



  j
j 0

t

s

[ F MO (t ; )]  ( j n)1 f MO (t ,  ) dt

s

[ G (t ) 1   G (t )]

 ( j  n ) 1

[ g (t ) [1   G (t )] 2 ] dt



m 1

   j s, 0,  ( j  n ) 1
j 0

 ( j  n ) 1 

or E (T s ) 

  t
l

l 0

s

[ F MO (t; ) ] l f MO (t , ) dt



 ( j  n ) 1 



  t
l

l 0

s

l

[G (t ) 1   G (t )] [ g (t ) [1   G (t )] 2 ] dt



 ( j  n ) 1





l

s, l , 0

l 0


q

r

Where p, q , r   t p [G (t ) 1   G (t )] [ G (t ) 1   G (t )] [ g (t ) [1   G (t )] 2 ] dt


is the PWM of MO ( ) distribution.
Therefore the moments of the BGMO - G(t ; , , m, n) can be expresses in terms of the PWMs
of MO ( ) (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.

19

Proceeding as above we can derive s th moment of the r th order statistic Tr:n , in a random
sample of size n from BGMO  G on using equation (17) as
 ( j  n ) 1 

E (T s r ;n ) 

 
l0

l, k

 s, k l, 0

k 0

Where  j ,  l and  l , k defined in above
4.5 Moment generating function
The moment generating function of BGMO  G 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

sT


st

M T ( s )  E [e ]   e f

BGMO



m 1



   j  e st
j 0



m1

(t ; , , m, n ) dt   e st   j
j 0



d
[ F MO (t;  )]  ( j n ) dt
dt

m 1

d
[ F MO (t ; )]  ( j  n ) dt    j M X ( s )
dt
j 0

Where M X (s ) 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



I R ( )  (1   ) 1 log  f (t )  dt 
 

where   0 and   1 For furthers details, see Song (2001). Using binomial expansion in (6) we
can write
f

BGMOG

(t ; , , m, n ) 

m 1
n 1
 1

   G (t )       G (t )    




g
(
t
)

G
(
t
)

 1 

 
 
 B (m, n) [1   G (t )] 2  [1   G (t )]   1   G (t )    1   G (t )   

 
 



f MO (t ,  )  [ F MO (t ,  )]  ( 1) [1  {F MO (t ,  )}  ]  ( m1) [ F MO (t , ) ]  ( n1)

B ( m, n )


f MO (t , )  [ F MO (t , )]  ( n 1) [1  {F MO (t , )}  ]  ( m 1)
B ( m, n ) 
 ( m 1)
  (m  1) 

MO

MO
 ( n 1)

 (1) j [ F MO (t ; )]  j

f
(
t
,

)
[
F
(
t
,

)]


j 
B ( m, n )
j 0 





f MO (t , ) 

B (m, n)

 ( m 1)


j 0

  (m  1) 

 (1) j [ F MO (t ; )]  j  ( n 1)
j 


20



Thus the Rényi entropy of T can be obtained as

  ( m 1)  MO

I R ( )  (1   ) log   Z j  f (t , )  [ F MO (t ; )]  j  ( n 1) dt 
 j 0 

1

Where Z j 


B ( m, n ) 

  (m  1) 

 (1) j
j



4.7 Quantile power series and random sample generation
The quantile function T, let t  Q (u )  F 1 (u ) , can be obtained by inverting (7). Let z  Qm , n (u ) be
the beta quantile function. Then,
1

  [1  {1  Q (u )}  ] 
m, n

t  Q (u )  Q G 
1
 1   [1  {1  Q (u )}  ] 
m,n


It is possible to obtain an expansion for Qm,n (u ) in the Wolfram website as
It is possible to obtain an expansion for Qm,n (u ) as


z  Qm, n (u )   e i u

i m

i0

(see

“Power

series”

From

MathWorld--A

Wolfram

Web

Resource.

http://mathworld.wolfram.com/PowerSeries.html)
Where e i  [ m B ( m, n) ] 1 m d i and d 0  0, d 1  1, d 2  (n  1) ( m  1) ,
d3 

(n  1) (m 2  3mn  m  5n  4)
2 (m  1) 2 (m  2)

d 4  (n  1) [ m 4  (6 n  1) m 3  (n  2) (8n  5) m 2  (33n 2  30n  4) m
 n (31n  47)  18 ] [ 3(m  1) 3 (m  2) (m  3) ]. . .
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 BGMO  G family these measures are given by
B

Q(3 4)  Q(1 4)  2Q(1 2)
Q (3 8)  Q (1 8)  Q(7 8)  Q(5 8)
and M 
Q(3 4)  Q(1 4)
Q(6 8)  Q (2 8)

For example, let the G be exponential distribution with parameter   0, having pdf and cdf as
g (t :  )   e  t , t  0 and G (t :  )  1  e  t , respectively. Then the p th quantile is obtained as

 (1 /  ) log [1  p ] . Therefore, the p th quantile t p , of BGMO  E is given by

21

1



 [1  {1  Qm , n ( p )} ] 
1


t p   log 1 
1

 1   [1  {1  Q ( p )}  ] 
m, n



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 t  0 are given by
f (t ) ~

 g (t )
[1  {  1   } ] m1
B(m, n) 

as G (t )  0

F (t ) ~

1
[1  {  1   } ] m
B ( m, n ) m

as G (t )  0

h (t ) ~

 g (t )
[1 {  1   } ] m 1
B(m, n) 

as G (t )  0

Proposition 2. The asymptotes of equations (6), (7) and (8) as t   are given by
f (t ) ~

1  F (t ) ~

  n g (t ) G (t ) n 1
B (m, n)

as t  

[ G (t )]  n
n B ( m, n )

as t  

h(t ) ~  g (t ) G (t ) 1

as t  

5. Estimation
5.1 Maximum likelihood method
The model parameters of the BGMO  G distribution can be estimated by maximum likelihood. Let
t  (t1 , t 2 ,...t n )T be a random sample of size n from BGMO  G with parameter
vector θ  ( m, n, , , βT ) T , where β  ( 1 ,  2 ,...  q ) T corresponds to the parameter vector of the
baseline distribution G. Then the log-likelihood function for θ is given by
r

r

  (θ)  r log  r  log    log [ g (ti , β) ]  (  1)  log [G (ti , β)]  r log[ B(m, n)]
i 0

r

i 0

r

 (  1) log [1   G (t i , β)]  (m  1) log [ 1  [  G (ti , β) {1   G (ti , β ) } ] ]
i 0

i 0

r

 (n  1) log [ [  G (t i , β) {1   G (t i , β) } ]  ]

(18)

i0

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).

22

By

taking

the

partial

derivatives

of

the

log-likelihood

function

with

respect

to

m, n,  , and β components of the score vector
U θ  (U m , U n , U ,U  ,U  T ) T can be obtained as follows:

Um 

r

 r (m)  r (m  n)   log [ 1  [ G (ti , β) {1   G (ti , β) }] ]
m
i 0

Un 

r

 r (n)  r (m  n)  b  log [[ G (ti , β) {1   G (ti , β) } ]  ]
n
i0

U 

r
r
 r
  r log    log [G (ti , β)]   log [1   G (ti , β)]
 
i 0
i 0

[  G (ti , β) {1   G (ti , β) } ]  log [ G (ti , β) {1   G (ti , β) }]
1  [  G (ti , β) {1   G (ti , β) } ]
i 0
r

 (1  m)

[ G (ti , β) {1   G (ti , β) } ]  log [ G (ti , β) {1   G (ti , β) } ]
[ G (ti , β) {1   G (ti , β) } ] 
i 0
r

 (n  1)
U 

r
 r 
G (ti , β)

 (  1)


i  0 1   G (ti , β )

r
G (ti , β)  G (ti , β)
1
G (ti , β)
  (n  1)



i  0 [{1   G (ti , β)}  { G (ti , β)} ][1   G (ti , β )]
i  0 1   G (t i , β )
r

   (1  m)

U 

r
r
r

g (β ) (ti , β)
G ( β ) (ti , β)
 G ( β ) (ti , β)

 (1   ) 
 (  1)
 β i0 g (ti , β)
G (ti , β)
i 0
i  0 1   G (ti , β)

G (ti , β)  1 G ( β ) (ti , β)


i 0 [ {1   G (t i , β )}  { G (ti , β)} ][1   G (ti , β)]
r

    (m  1)
r

  (1  n)
i 0

G ( β ) (ti , β)
[1   G (ti , β)]G (ti , β)

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 loglikelihood function with respect to each parameter. The i j th elements of I n (θ) are given by

  2 l (θ) 
 ,
I i j  E 
   i  j 



i, j  1, 2, , 3  q

The exact evaluation of the above expectations may be cumbersome. In practice one can estimate
I n (θ) by the observed Fisher’s information matrix Î n (θˆ ) is defined as:

23

  2 l (θ) 

Î i j   
    
i
j 


,

i, j  1, 2,, 3  q

θ  θˆ

Using the general theory of MLEs under some regularity conditions on the parameters as n   the
1
n (θˆ  θ) is N k (0,Vn ) where Vn  (v jj )  I n (θ) . The asymptotic

asymptotic distribution of
behaviour remains valid if

Vn

Vˆ  Î 1 (θˆ )
is replaced by n
. 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 (1   / 2) 100% confidence interval for the mle of jth parameter  j are
respectively given by

v̂ j j and ˆ j  Z  / 2 vˆ j j , where Z  / 2 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 BGMO  E (m, n, , ,  ) 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 BGMO  G
distribution can be a better model than GMOKw  G (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 AIC  2 k  2 l ;

BIC  k log (n)  2 l ; CAIC  AIC 

2 k (k  1)
; and HQIC  2 k log [log (n)]  2l
n  k 1

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}

24

Table 1: MLEs, standard errors and 95% confidence intervals (in parentheses) and
the AIC, BIC, CAIC and HQIC values for the data set.
Parameters

â

b̂

̂

ˆ

̂
ˆ
log-likelihood( lmax )
AIC
BIC
CAIC
HQIC

GMOKw  W

BGMO  W

1.178
(0.017)
(1.14, 1.21)
0.291
(0.209)
(-0.12, 0.70)
0.617
(0.002)
(0.61, 0.62)
1.855
(0.003)
(1.85, 1.86)
1.619
(0.977)
(-0.29, 3.53)
0.178
(0.144)
(-0.10, 0.46)
-90.99

1.187
(0.702)
(-0.19, 2.56)
2.057
(2.240)
(-2.33, 6.45)
0.009
(0.006)
(-0.00276, 0.02)
4.194
(0.668)
(2.88, 5.50)
0.047
(0.108)
(-0.16, 0.26)
0.017
(0.016)
(-0.01, 0.05)
-80.38

193.98
204.11
196.53
197.65

172.76
182.89
175.31
176.43

Estimated cdf's
1.0

Estimated pdf's
BGMO-W
GMOKw-W

0.6
0.4

cdf

0.15

0.0

0.00

0.05

0.2

0.10

Density

0.20

0.8

0.25

BGMO-W
GMOKw-W

0

2

4

6

8

10

0

2

4

6

X

X

(a)

(b)

8

Fig: 3 Plots of the observed (a) histogram and estimated pdf’s and (b) estimated cdf’s for the
BGMO  G and GMOKw  G for example I.

25

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}

26

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

GMOKw  W

BGMO  W

0.765
(0.025)
(0.72, 0.81)
2.139
(0.774)
(0.62, 3.66)
4.271
(0.018)
(4.24, 4.31)
2.919
(0.013)
(2.89, 2.94)
1.097
(0.309)
(0.49, 1.70)
0.114
(0.042)
(0.03, 0.19)
-111.75

0.866
(0.159)
(0.55, 1.18)
0.329
(0.167)
(0.00168, 0.66)
2.131
(1.182)
(-0.19, 4.45)
2.223
(0.725)
(0.80, 3.64)
3.285
(3.901)
(-4.36, 10.93)
2.635
(1.310)
(0.07, 5.20)
-109.28

235.50
258.58
235.75
244.69

230.56
253.64
230.80
239.76

â

b̂

̂

ˆ

̂
ˆ
log-likelihood( lmax )
AIC
BIC
CAIC
HQIC

1.0

Estimated cdf's

1.5

Estimated pdf's
BGMO-W
GMOKw-W

0.0

0.0

0.2

0.5

0.4

cdf

Density

0.6

1.0

0.8

BGMO-W
GMOKw-W

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

X

X

(a)

(b)

1.5

2.0

Fig: 4 Plots of the observed (a) histogram and estimated pdf’s and (b) estimated cdf’s for the
BGMO  G and GMOKw  G for example II.

27

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

â

b̂

̂
ˆ

̂

ˆ
log-likelihood( lmax )
AIC
BIC
CAIC
HQIC

GMOKw  W

BGMO  W

1.015
(0.071)
(0.88, 1.15)
0.385
(0.168)
(0.06, 0.71)
0.803
(0.003)
(0.79, 0.81)
2.222
(0.004)
(2.21, 2.23)
1.482
(0.440)
(0.62, 2.34)
0.345
(0.169)
(0.01, 0.68)
142.63

1.458
(1.123)
(-0.74, 3.66)
0.734
(1.810)
(-2.81, 4.28)
0.598
(1.496)
(-2.33, 3.53)
2.439
(0.779)
(0.91, 3.97)
0.685
(1.150)
(-1.57, 2.94)
0.201
(0.573)
(-0.92, 1.32)
-141.29

297.26
312.89
298.16
303.59

294.58
310.21
295.48
300.92

28

Estimated cdf's
1.0

Estimated pdf's
BGMO-W
GMOKw-W

0.6
0.4

cdf

0.3

0.0

0.0

0.1

0.2

0.2

Density

0.4

0.8

0.5

BGMO-W
GMOKw-W

0

1

2

3

4

5

6

0

1

2

X

3

4

5

6

X

(a)
(b)
Fig: 5 Plots of the observed (a) histogram and estimated pdf’s and (b) estimated cdf’s for the
BGMO  G and GMOKw  G 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 BGMO  W distribution turns out to be a
better distribution than GMOKw  W 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

BGMO  E

The pdf of the BGMO  E distribution is given by
1     e  t [ e  t ]  1    e  t 
f BGMOE (t ) 
1 
B (m, n) [1   e t ]  1  1   e t 


29







m 1

   e  t   


 t 
 1   e  

n 1

For a random sample of size n from this distribution, the log-likelihood function for the parameter
vector θ  (m, n, , ,  ) T is given by
r

r

  (θ)  r log  r  log   r log     ti  (  1)  log (e  t i )  r log[ B(m, n)]
i 0

r

i0

r

 (  1) log [1   e ti ]  (m  1) log [1  [  e ti {1   e ti } ]  ]
i 0

i 0

r

 (n  1) log [  e ti {1   e ti } ] 
i 0

The components of the score vector θ  (m, n, , ,  ) T are
Um 

r

 r (m)  r (m  n )   log [1  [  e  t i {1   e  t i } ]  ]
m
i 0

Un 

r

 r (n)  r (m  n)   log [  e  ti {1   e  ti }] 
n
i0

U 

r
r
 r
  r log    log [e  ti ]   log [1   e  t i ]
 
i 0
i 0

[ e  ti {1   e  ti } ]  log [ e  t i {1   e  t i }]
 (1  m)
1  [ e  ti {1   e  ti } ]
i 0
r

[  e  t i {1   e  t i } ]  log [  e  t i {1   e  t i } ]
 (n  1)
[  e  ti {1   e  t i } ] 
i 0
r

U 

r
 r 
e  ti

 (  1)
 t i


i0 1   e
r
[e  t i ]  [1  e  t i ]
1
1  e  ti


(
n

1
)

 t i 
 t i
}  { e  t i }  ] [1   e  ti ] 
i  0 [{1   e
i 0 1   e
r

   (1  m)

U 

r
r
r

r
 e  t i
  e  ti
   ti  (  1) 

(


1
)

 t i
 t i
   i 0
i0 1  e
i 0 1   e

[e ti ] 
 ti 
}  { e ti }  ][1   e ti ]
i  0 [ {1   e
r

     (m  1)
r

   (1  n)
i 0

e ti
[1   e ti ] e ti

The asymptotic variance covariance matrix for mles of the unknown parameters θ  (m, n, ,  ,  )
of BGMO  E (m, n,  ,  ,  ) distribution is estimated by

30

 var (mˆ )

 cov ( nˆ , mˆ )

Î n 1 (θˆ )   cov (θ̂, mˆ )

 cov ( α̂, mˆ )

 cov ( λ̂, mˆ )


cov ( mˆ , nˆ ) cov ( mˆ , θ̂)

cov ( mˆ , α̂)

var ( nˆ )

cov ( nˆ , θ̂)

cov ( nˆ , α̂)

cov (θ̂, nˆ )

var (θ̂ )

cov (θ̂, α̂ )

cov (α̂ , nˆ )

cov (α̂, θ̂ )

var (α̂ )

cov (λ̂, nˆ )

cov ( λ̂, θ̂ )

cov ( λ̂, α̂ )

cov ( mˆ , λ̂) 
cov ( nˆ , λ̂) 

cov (θ̂, λ̂) 

cov (α̂, λ̂ ) 

var ( λ̂) 


   2 l (θ )  

Where the elements of the information matrix Iˆ n (θˆ )    
   i   j  

 
the following second partial derivatives:
 2
  r (m)  r (m  n)
m2

can be derived using
θ  θˆ

 2
  r (n)  r (m  n)
n2
r

r
[ α e  λti {1  α e  λti } ] 2 θ log [α e  λti {1  α e  λti }] 2
 2
  2  (1  m)
θ
θ 2
[1  [ α e  λti {1  α e  λti } ]θ ] 2
i 0


r

 (1  m)



[ α e  λti {1  α e  λti } ] θ log [α e  λti {1  α e  λti }] 2
1  [ α e  λti {1  α e  λt i } ]θ

i 0
r

e  2 λti
 2 r θ
 (θ  1)

 λt i 2
α 2 α 2
)
i  0 (1  α e



r

 θ (n  1)



e λti [1  αe  λti ] - 2 α e 3 λti ti {1  α e  λti } 3 - 2 α e  2 λti ti {1  α e  λti } 2
b
i 0





r

 θ (n  1)

- α e  2 λti ti {1  α e  λti } 2  α e  λti ti {1  α e  λti }
b
i 0


r

 θ (n  1)





e λti [1  αe  λti ] - α e  2 λti ti {1  α e  λti } 2  α e  λti ti {1  α e  λti }
b

i 0



2

θ [ α e  λti {1  α e  λti }] θ -1  2 α e 3 λti {1  α e  λti } 3 - 2 e  2 λti {1  α e  λti } 2 


 (1  m)
 λt i
 λt i θ
1 [αe
{1  α e
}]
i 0
r


r

 (1  m)


i 0

r

 (1  m)


i 0



θ 2 [ α e λti {1  α e  λti } ] 2 θ-2  α e 2 λti {1  α e λti } 2  e  λti {1  α e  λti }



2

[1  [ α e  λti {1  α e λti } ]θ ] 2



θ (θ - 1)[ α e  λti {1  α e  λti }] θ -2  α e  2 λti {1  α e  λti } 2  e  λti {1  α e  λti }
1  [ α e  λti {1  α e  λti }]θ

31



2

r
 2
  2  (θ  1)
2

λ



i 0







e  λti [1  α e  λti ]ti - α α e  2 λti ti {1  α e  λti } 2  e  λti ti {1  α e  λti }
b
e

r

 θ (n  1)



α ti - αα e  2 λti ti {1  α e  λti } 2  e  λti ti {1  α e  λti }
b
i 0
r

 θ (n  1)

 α 2 λ e  2 λti ti 2 α λ e  λti ti 2 



 λt i 
 [1  α e  λti ] 2
1

α
e
i 0 




r

 θ (n  1)

r

 λt i

[1  α e


i 0

 λt i

 - 2 α 2α e 3 λti ti 2 {1  α e  λti } 3  3 α α e  2 λti ti 2 {1  α e  λti } 2 

]ti 
 λt i 2
 λt i 

 α e ti {1  α e
}

b



r

θ 2 [ α e  λti {1  α e  λti } ] 2θ -2 - α α e  2 λti ti {1  α e  λt i } 2  αe  λti {1  α e  λti }
 (1  m)
[1  [ α e  λti {1  α e  λti } ]θ ] 2
i 0





2



r

θ (θ - 1) [ α e  λti {1  α e  λti } ] θ -2 - α α e  2 λti ti {1  α e  λti } 2  αe  λti {1  α e  λti }
 (1  m)
1  [ α e  λti {1  α e  λti } ]θ
i 0



θ [αe

r

 (1  m)

 λt i

{1  α e

 λt i



2

α e 3 λti ti 2 {1  α e  λti } 3  3 α α e  2 λti ti 2 {1  α e  λti } 2 
 α e  λti ti 2 {1  α e  λti } 
1  [ α e  λti {1  α e  λti } ]θ



}]



θ -1  - 2 α




i 0

2

 2
 r (m  n)
mn
r

[ α e  λti {1  α e  λti } ] θ log [α e  λti {1  α e  λti }]
 2

 mθ
1  [ α e  λti {1  α e  λti } ]θ
i 0



 2
 m α

r









θ [ α e  λti {1  α e  λti } ] θ -1  [ α e  2 λti {1  α e  λti } 2 ]  [ α e  λti {1  α e  λti } ]

i 0

1[αe

r



 λt i

{1  α e

 λt i

θ

}]



 2
θ [ α e  λti {1  α e  λti } ] θ -1  [ α α e  2 λti ti {1  α e  λti } 2 ]  [ α e  λti ti {1  α e  λti } ]

 m 
1  [ α e  λti {1  α e  λti } ]θ
i 0



 2

 nθ

r

 log[ α e  λt ti
i

{1  α e  λti } ]

i 0
r



e  λti [1  α e  λti ] - α e  2 λti {1  α e  λti } 2  e  λti {1  α e  λti }
 2
θ
 nα
b
i 0



32





r

 2
e  λti [1  α e  λti ] - α α e  2 λti ti {1  α e  λti } 2  α e  λti ti {1  α e  λti }
θ
 n λ
b
i0



 2
r
 
θα α
r

 (n  1)

e



r

r

 (1  m)



e  λti

 1  α eλt

i 0
λt i

i



[1  αe  λti ] - α e  2 λti {1  α e  λti } 2  e  λti {1  α e  λti }
b

i 0





[ α e  λti {1  α e  λti } ] θ -1 - α e  2 λti {1  α e  λti } 2  e  λti {1  α e  λti }
1 [αe

i 0

 λt i

{1  α e

 λt i



θ

}]



r

 θ(1 - m)





[ α e  λti {1  α e λti } ] 2 θ-1 - α e  2 λti {1  α e  λti } 2  e  λti {1  α e λti } log[ e  λti {1  α e  λti }]
[1  [ α e  λti {1  α e  λti } ]θ ] 2
i 0





r

 θ(1 - m)



[ α e λti {1  α e λti } ] θ -1 - α e 2 λti {1  α e λti } 2  e  λti {1  α e λti } log[ e  λti {1  α e λti }]



1[ α e

i0

r

r





 λ ti

 λ ti

{1  α e

θ

}]

 2
ti  log[ α e  λti ti {1  α e  λti } ]

 θ λ
i 0
i 0



r

 (n  1)

e  λti [1  α e  λti ] - α α e  2 λti ti {1  α e  λti } 2  α e  λti ti {1  α e  λti }
b
i 0


r

 (1  m)





[ α e  λti {1  α e  λti } ] θ -1 - αα e  2 λti ti {1  α e  λti } 2  e  λti {1  α e  λti }
1 [αe

i 0

 λt i

{1  α e

 λt i



θ

}]



r

 θ(1 - m)





[ α e λti {1  α e λti }] 2 θ-1 - α α e 2 λti {1  α e  λti ti } 2  e  λti ti {1  α e λti } log[ e  λti {1  α e λti }]
[1  [ α e  λti {1  α e λti } ]θ ] 2
i 0


r



 θ(1 - m)



1 [ α e

i0

 2
 (θ  1)
  
r

 θ (n  1)


i 0
r

 θ (n  1)



[ α e  λti {1  α e  λt i } ] θ -1 - α α e  2 λti ti {1  α e  λti } 2  e  λt i ti {1  α e  λti } log[ e  λti {1  α e  λti }]

r

 2 λt i

ti
e  λti ti
 α e

 λt i 2

]
1  α e  λti
i  0  [1  α e



 λt i

{1  α e

 λt i

θ

}]










ti - α e  2 λti {1  α e  λti } 2  e  λti {1  α e  λti }
b





e  λti [1  α e  λti ]ti - α e  2 λti {1  α e  λti } 2  e  λti {1  α e  λti }
b
i 0



33

 2 α αe 3 λti ti {1  α e  λti } 3  α e  2λti ti {1  α e  λti } 2


e  λti [1  α e  λti ] 
 2 λt i
 λt i 2
 λt i
 λt i 
r

 2α e
ti {1  α e }  e ti {1  α e
}

 θ (n  1)
b
i 0


r

 (1  m)

θ {αe

 λt i

(1  α e



 λt i

)}

θ 1

 2 α αe  3 λti ti {1  α e  λti } 3  α e  2λti ti {1  α e  λti } 2



 2 λt i
 λt i 2
 λt i
 λt i 

 2α e
ti {1  α e }  e ti {1  α e
}

1  {α e  λti (1  α e  λti )} θ

i 0

θ 2 {αe  λti (1  α e  λti )} 2θ  2 (α e  2 λti {1  α e  λti } 2  e  λti {1  α e  λti })



r

 (1  m)

  (α αe  2 λti ti {1  α e  λti } 2  α e  λti ti {1  α e  λti })





[1  {α e  λti (1  α e  λti )} θ ] 2

i 0

θ ( θ - 1) {αe  λti (1  α e  λti )} θ  2 (α e  2 λti {1  α e  λti } 2  e  λti {1  α e  λti })
r

 (1  m)


i 0



  ( α αe  2 λti ti {1  α e  λti } 2  α e  λt i ti {1  α e  λti })
1  {α e

 λt i

(1  α e

 λt i



)} θ

Where  (.) is the derivative of the digamma function.

References
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