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IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728. Volume 5, Issue 2 (Jan. - Feb. 2013), PP 34-40
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On new method of estimation of parameters of Size-biased
Generalized Gamma Distribution and its Structural properties
A.Ahmed1, K.A. Mir2 , J. A. Reshi1
2
Department of Statistics, Govt. Degree College Bemina, Srinagar Kashmir, India.
1
Department of Statistics, University of Kashmir, India.
Abstract: In this paper, a new class of size-biased Generalized Gamma distribution is introduced. The estimates
of parameters of size-biased Generalized Gamma distribution are obtained by using new method of moments. A
new distribution which contains as a special case is introduced. The characterizing properties of the model are
derived. Also, other important property including Shannon entropy which is the measure of the uncertainty in
this class of the distribution are derived and studied.
Key words: Generalized Gamma distribution, Size-biased Generalized Gamma distribution, moment generating
function, sample coefficient of variation, Shannon’s entropy
I. Introduction
The Generalized gamma distribution presents a flexible family in the varieties of shapes and hazard
functions for modelling duration. It was introduced by Stacy [1]. Distributions that are used in duration analysis
in economics include exponential [2, 3], lognormal [4], gamma [5], and Weibull [6]. The generalized gamma
distribution, which encompasses exponential, gamma, and Weibull as subfamilies, and lognormal as a limiting
distribution, has been used in economics by Jasggia [7]. In information theory, the important properties
including Shannon entropy and Fisher’s information matrix which are the measure of the uncertainty in this
class of the distribution are derived and studied. The concept of Shannon entropy [8] is the central role of
information theory, some-times referred as measure of uncertainty. The entropy of a random variable is defined
in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty.
Although Prentice [9] have presented a procedure to obtain the three parameters of the generalized gamma
distribution, his procedure still quit complicated. In this paper, we propose a simple procedure to obtained three
estimators by using its characterization and moment estimation approach. Note that Hwang .T and Huang. P
[10] have obtained more general characterizations with the independence of sample coefficient of variation
Vn
with sample mean X n as one of its special cases when random samples are drawn from the generalized gamma
2
distribution. Their characterization is used to derive the expectation and the variance of Vn and then the new
estimators for the three parameters of size-biased generalized gamma distribution are proposed.
The pdf of the generalized gamma distribution is given by:
f ( x;  ,  , k ) 

k

(x) k 1 e ( x ) , for x  0 and  ,  , k  0
(1)
The mth non-central moments of GGD is given by

m
 k  

E( X m )   m
 k
m 1,2,........
(2)
II.
Size biased Generalized Gamma Distribution
c
A size- biased Generalized Gamma distribution (SBGGMD) is obtained by applying the weights x ,
where c =1 to the Generalized Gamma distribution.

f S ( x;  ,  , k )  
0
x f ( x;  ,  , k )
dx
E( X )
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On new method of estimation of parameters of Size-biased Generalized Gamma Distribution and its

f S ( x;  ,  , k )  
x

x k 1 e  ( x )
k

dx

1
k  





 ( k )
0

f S ( x;  ,  , k )  


(x) k e ( x ) dx

1
 k  


Where g w ( x;  ,  , k ) represents a probability density function. This gives the size -biased generalized
0
gamma distribution (SBGGMD) and its pdf is given by

f S ( x;  ,  , k ) 
(x) k e ( x )


1
 k  


For   0, k  0 ,   0 and k  1
(3)
2.1
Special cases:
1.
When   k  1, then Size biased Generalized Gamma distribution reduced to Size-biased
exponential distribution and its probability distribution is given by
f S ( x;  )  2 x e x
,  0
(4)
2.
When   1 then Size biased Generalized Gamma distribution reduced to size-biased gamma
distribution and its probability distribution is given by
k 1
e x x k ,   0 , k  0 ,0  x  
(5)
k  1
When k  0 and   1 then Size biased Generalized Gamma distribution reduced to exponential
f S ( x;  , k  1) 
3.
distribution and its probability distribution is given by
f S ( x;  )   e x
,   0 ,0  x  
(6)
Theorem 2.2. Let X  ( x1 , x2 , x3 ...xn ) be an identical independently distributed sample from size-biased
generalized gamma distribution

f S ( x;  ,  , k ) 
(x) k e ( x )

1
 k  



m 1

  k 
 

m
then, E ( X ) 

1
m  k  



m  1,2,........
Proof: Using equation (2), the mth non central moment is given by,

E( X m ) 
x
0
E( X ) 
m

m

1
 k  


 k 1


(x) k e ( x ) dx

( k  r 1)1 ( x )
e
dx
x

1
 k   0


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On new method of estimation of parameters of Size-biased Generalized Gamma Distribution and its
On solving the above equation, we get

m `1 

 

m
E( X ) 

1
m  k  


  k 
III.
(7)
New Moment Estimator Of The Size-Biased Generalized Gamma Distribution
For deriving new moment estimators of three parameters of the size-biased generalized gamma
distribution, we need the following theorem obtained by using the similar approach of Hwang .T and Huang .P
(Theorems of 2006).
Theorem 3.1. Let
n  3 and let X 1 , X 2 , X 3 ... X n be a n positive identical independently distributed random
variables having a probability density function
sample coefficient of variation Vn 
density where
f (x) .Then the independence of the sample mean X n and the
Sn
is equivalent to that f (x) is a size-biased generalized gamma
Xn
S n is the sample standard deviation.
2
S 
The next theorem is easy to prove and need to derive the expectation and the variance of Vn   n  , where
 Xn 
X n and S n are respectively the sample mean and the sample standard deviation.
2
Theorem 3.2: Let
n  3 and let X 1 , X 2 , X 3 ... X n be a n positive identical independently distributed random
samples drawn from a population having a size-biased generalized gamma density
f S ( x;  ,  , k ) 



1
 k  


 
Then
E (S n ) 
2
  k 
 
(x) k e ( x ) , x  0,   0,   0, k  0
1 
3
2 

 k      k  
  

 


2  2  k 

1

 
Proof :

m `1 

 
Here,

m
E( X ) 
, m  1,2,3,...

1
m  k  


  k 

2




E( X n ) 

1 
k 

 



 
k 


E Xn
2

k 
 

3  
1 

2 
 
K 
  ( n  1)  2  2 
k 

 
 
 
 




1
nk  2 
k  





And E ( S n )  n.V ( X n )
2
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On new method of estimation of parameters of Size-biased Generalized Gamma Distribution and its
 
E (S n ) 
2
  k 
 

1 
3
2 
 k      k  
 

 


1
2  2  k  


(8)
2
Where X n and S n are respectively their sample mean and sample variance.
Theorem 3.3: Let n  3 and let X 1 , X 2 , X 3 ... X n be a n positive identical independently distributed random
samples drawn from a population having a size-biased generalized gamma density

f S ( x;  ,  , k ) 

(x) k e (x ) , x  0,   0,   0, k  0
Then
 1
 k  
 
 

3 
1
2 
n  k   k     2  k  
2
S 
 

 


E  n   
 X n   k  3  k  1   n  1 2  k  2 


  
 
 


2
Where X n and S n are respectively their sample mean and sample variance.
Proof: By theorem 3.1, we have
 Sn 2
2
E ( S n )  E  2 . X n  
 Xn

2
And hence
 
 Sn 2 
2
E  2 ..E X n
 Xn 
 
 
 Sn 2  E Sn 2
E  2  
2
 Xn  E Xn
Applying theorem 3.2 to the above identity yields that
 

3 
1
2 
n  k   k     2  k  
S 
 

 


E  n   
 X n   k  3  k  1   n  1 2  k  2 


  
 
 


2
(9)
Thus 3.3 is established.
 
3 
1 
 k   k  
S 
 
 

E  n   
 1 as n   and that this limit is the square of
X


2
2
n


  k  


2
Note that
2
the population coefficient of variation. Thus,
Sn
is an asymptotically unbiased estimator of the square of the
2
Xn
population coefficient of variation.
Theorem 3.4: Let n  3 and let X 1 , X 2 , X 3 ... X n be a n positive identical independently distributed random
samples drawn from a population having a size-biased generalized gamma density
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On new method of estimation of parameters of Size-biased Generalized Gamma Distribution and its

f S ( x;  ,  , k ) 


1
 k  


 
k 1

E e tx 
Then


   e
Ee
Proof: Here

tx 
(x) k e ( x ) , x  0,   0,   0, k  0
t
tx 

k
1

f x;  ,  , k dx
0
 


E e tx   e tx
0
 
k 1



1
 k  


k 1

E e tx 
x k e x  dx

 
x k e  x  t dx
1

 k   0


1
dt
x  1
x  0, t  0 and x  , t  
Put x
 
Ee
 

 

t
1
k

 k   0


 
E e tx
 t x  t
k 1
tx 
Ee

k 1
tx 

 k   0



1



k


 
k 1


1
k

1 
 k     t 


Substitute
1
t



e t  t dt

k 1

E e tx 
dx 

dt
e t  t  1
 k  1 1
t
1 




t

k
1

  1 in the above relation, we have
k 1
E e tx  
  t k 1
(10)
(11)
Shannon’s entropy of size-biased Generalized Gamma Distribution
For deriving be entropy of the size-biased Generalized Gamma distribution, we need the following two
definitions that more details of them can be found in [11].
Definition 3.2.1: The entropy of the discrete alphabet random variable f defined on the probability space is
defined by:
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On new method of estimation of parameters of Size-biased Generalized Gamma Distribution and its
n
H P ( f )   p( f  a) log( p( f  a))
i 1
It is obvious that
HP ( f )  0
Definition 3.2.1: The oblivious generalizations of the definition of entropy for a probability density function f
defined on the real line as:

H ( f )    f ( x) log f ( x)  E ( log( x))
0
Provided the integral exists.
Theorem 3.5: Let X 1 , X 2 , X 3 ... X n be a n positive identical independently distributed random samples drawn
from a population having a size-biased generalized gamma density
f S ( x;  ,  , k ) 



1
 k  


(x) k e ( x ) , x  0,   0,   0, k  0
Then Shannon’s entropy of Size-biased Generalized Gamma Distribution is:



2


 k  
k 1


1
 
  
H  f x;  , k ,     log
 k  K    log      




1
1

  k     
 k  
 


 
Proof: Shannon’s entropy is defined as:
H  f ( x; ,  , k )  E log f ( x; ,  , k




 k 1

H  f ( x; ,  , k )  E  log 

1
 


k  





x k e x 

















 k 1

H  f ( x; ,  , k )   log 

1
 


k  


 











k

E
log
x



E
(
x
)













 k 1
H  f ( x; ,  , k )   log 

  k  1




 








  k E log x     E ( x)








 k 1
H  f  x;  , k ,     log 

1
  k 


 


2

 k  




 k E log x    


1
 
 k  



(12)

Now,
E (log( x))   log xf ( x; ,  , k )dx
0

E (log( x)) 
 log x
0
k 1

1
k 






x k e x  dx

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(13)
39 | Page
On new method of estimation of parameters of Size-biased Generalized Gamma Distribution and its
1
Let
x 

 t 
 t x    x    

 

t
k 1

1

1
dx 
dt
 t
1
1

k
 t    t  t dt
E (log( x)) 
log
    e
1
1

1  0      

 k  
 t


1
 t   k   1 t
E (log( x)) 
log
e dt
  t

1  0   
 k  


1
1



k  1
k  1

1
t

 log t e t t  dt   log 
E (log( x)) 
e
t
dt
0




1  0 
1
  k  
  k  


 



1


1

log  
k   

k 

 

E (log( x)) 


1
1
  k  
 k 





1

 





1
 k    log 

E (log( x))  

(14)
Substitute the value of equation (14) in equation (12), we have



2


 k  
k 1

1
 
  



H  f x;  , k ,     log

k

K


log













1 
1

  k     
 k  
 


 
(15)
The relation (15) is the Shannon’s entropy of Size-biased Generalized Gamma Distribution.
References
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[2].
[3].
[4].
[5].
[6] .
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[8] .
[9] .
[10].
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40 | Page