Moments of Skew Normal Vector Quadratic Froms

Vol. 32, 2015, pp. 13-20

A Comparative Study on Two Risks for the Shape Parameter of Generalized-Exponential Distribution

Chandan Kumer Podder

Department of Statistics, University of Chittagong, Chittagong

Abstract

In this paper, a comparative study on two risk functions for Modified Linear Exponential (MLINEX) and Squared Error (SE) loss functions using the shape parameter of Generalized- Exponential (GE) distribution have been presented.

1. Introduction

The Generalized-Exponential (GE) distribution introduced by Gupta and Kundu (1999) is an important lifetime distribution in survival analysis. This distribution can be used quite effectively in situations where a skewed distribution is needed.

Without any loss of generality, the probability density function of Generalized-Exponential (GE) distribution with shape parameter



and scale parameter

  1

(say), is given as

 ^{x ; }  ^{  e}

^

 ^{1  e}

^



^1

^{; x  0 ,   0}

f

^{x x . (1.1)}

It can be seen that at

  1

, the model (1.1) reduces to standard exponential distribution. The various properties and estimation of Generalized-Exponential (GE) distribution model have been studied by (Gupta and Kundu (2001a), Raqab and Ahsanullah (2001) and Raqab (2002)). However, Madi and Raqab (2007), Singh et. al. (2008) discussed the estimation and prediction in Bayesian approach of this distribution model.

The purpose of this paper is to study and compare the risk functions for the shape parameter of Generalized-Exponential (GE) distribution using MLINEX and SE loss functions.

2. Preliminary Theory

Let

X

be a random variable whose distribution depends on parameter



and let



denotes the parameter space of possible values of



. Now consider the general problem of estimating the unknown parameter



, from the results of a random sample of

n

observations. Denoting the sample observations

x

n

x

x

₁

,

₂

,  ,

by

x

^{, let}

 ^ˆ

be an estimate of



and also let

^L   ^{ ˆ , }

be the loss incurred by taking the value of the parameter



^{to be}

 ^ˆ

^.

If

   

be the prior density of



, then according to Bayes’ theorem the posterior density of



^{will be}

 x      p x

l  |  

, where

l   | x 

is the likelihood function of



given the sample

x

^{, and}

      



 l  x   d  x

p |

.

It follows that, for a given

x

, the expected loss i. e., risk of the estimator

 ^ˆ   x

^is

 ^ ˆ   x , ^  E



 L  ^ ˆ   x , ^  

R 

    ^{   }

  



    



 d

x p

x x l

L |

ˆ ,

_{. (2.1)}

Assume the existence of (2.1) and that sufficient regularities conditions prevail to permit differentiation under the integral sign, the optimum estimator

 ^ˆ   x

^of



will be a solution of the equation

   





 0 ˆ  |   





 L l x d

. (2.2)

The validity of (2.2) and the desirability that should lead to a unique solution necessarily impose restrictions of one’s choice of loss function and prior density of



^.

Here, the following loss functions are considered.

i) 1

  ^{ˆ , ˆ ln ˆ 1 ;}  ^{0 ,}  ⁰

















 







 



 

 





 



   



 



 







L

. (2.3)

ii)

L

2

    ^{ˆ ,}   c  ^ˆ   

²

^; c  ⁰

^. (2.4)

The loss functions

L

₁ is a modified linear-exponential (MLINEX) which an asymmetric and

L

₂ is a squared error and symmetric.

3. Main Results

Let us consider the case of estimating the single parameter



of the Generalized-Exponential (GE) distribution in the model (1.1). The likelihood function of (1.1) is given by

  _     ^{ }





 











^

 

n

i

e x n

x

n

i

xi i

i

e e

e x

l

1

1 1 log

1

1

1

|







=

 

ⁿ

e

^{T, (3.1)}

15

where

_    



 







n

i

x

x_{i i}

e e

1

1

1



and

_  



 





n

i

x_i

e T

1

1

1

log

^.

The maximum likelihood estimator of



_is

T

n

, where

T

is defined above. It is noted that the part of the

likelihood function which is relevant to Bayesian inference on the unknown parameter



_is



ⁿ

e

^T.

A mathematically convenient prior density for the problem under consideration is conjugate prior for the shape parameter



^{given by}

   

^



^1

;   0 ,   0 ,   0

 e

^{  , (3.2)}

is simply a member of the gamma family of distributions.

By combining equations (3.1) and (3.2), we obtain the posterior distribution of



   

 

 

0 , 0 ,

0

;

|

¹

  





 

^{   }









 

 



^{ }



n T n

n e

x T

, (3.3)

which is distributed as gamma distribution with parameters

 n   

and

 T   

.

The advantage of taking the prior distribution to be conjugate lies in the fact that the likelihood function

 x 

l  |

, the prior density

   

and the posterior density

   | x 

are all of the same functional form, thus ensuring mathematical tractability.

For the limiting case, when

    0

in (3.2), the subclass of prior density is given by

  1 ; 0



 

 



, (3.4)

which is a uniform density function. The particular form of (3.4) is precisely the prior density advocated by Jeffrey’s [3] when one is completely ignorant of the values of



, of course, from their admissible ranges.

Substitution from (3.4) and (3.1) in (2.2), the optimum estimator

 ^ˆ

^of



is a solution of

 











0

1

0

ˆ exp   





 L T

_n

d

, (3.5)

which follows that

 ⁿ  ^T

n 1

ˆ

1



  





 







 

.

Hence the optimum estimator

 ^ˆ

for the loss function (2.3) is given by

T K

B



 ˆ

_{, (3.6)}

where

 





1

 



 







  n

K n

and

_  



 





n

i

x_i

e T

1

1

1

log

.

Again, for the loss function given by (2.4), it follows from (3.5) that the optimum estimator

 ^ˆ

is given by

 

^{ }

 























0

1 0

1 1

exp exp ˆ















d T

d T

n n

S ,

also follows that

T n

S



 ˆ

_{, (3.7)}

which is same as the mean of the posterior distribution (3.3) when

    0

.

Since

x

is a Generalized-Exponential (GE) variate with parameter



^and

n

, then

 





 





n

i

x_i

e T

1

1 1

log is distributed as gamma distribution with parameters

n

^and



, i. e.,

 ^{, } 

~ G n

T

^.

The probability density function of

T

is

 ^;  ^exp  ^ 

¹

^{;  0 ,  0}

   

^



 T T T

T n

p

ⁿ

n

. (3.8)

Now we are interested in finding the risk functions for the estimators

 ^ˆ

_B ^and

 ˆ

_S with respect to MLINEX and SE loss functions considered in (2.3) and (2.4).

17

 ^ ˆ , ^ 



  ^ ˆ , ^  

B B

ML

E L

R 

    _





 

   

 1 ˆ ln ˆ ln 1









 



_

E

_B^

E

_B ^, (3.9)

which gives

  ^{ } _





 





 





 

n n n

R

_{ML B}

n  





 ˆ , ln

_{. (3.10)}

Similarly, the risk function of the estimator

 ˆ

_S with respect to MLINEX loss function is

  ^{ } _





 

 



 

 



  ln 1

ˆ ,

n n n n

n n

R

_{ML S}

  







^{ , (3.11)}

where

  











0

exp

1

ln y y y dy

n

ⁿ is the first differentiation of

 n

with respect to

n

.

The risk functions

R

ML

 ^{ ˆ}

B

^{, } 

and

R

ML

 ^{ ˆ}

S

^{, } 

in (3.10) and (3.11) involving the expression of gamma and di-gamma

  

functions are complicated.

The risk functions of the estimators

 ^ˆ

_B ^and

 ˆ

_S with respect to SE loss function are as follow;

 ^{ˆ ,}  ^

²



 



 

 



 T

E K

R

_{S B}

=

     

2 2

2

2

1

1 1 2

1

1 

















 



 

 

 



 n n K

K n

^{. (3.12)}

 ^{ ˆ}

S

^{, }   ^{  ˆ}

S

^{ } 

²

S

E

R

=

 

  

2

2 1

2 





 n n

n

. (3.13)

The both risk functions (3.12) and (3.13) under squared error loss function are quadratic in



and it is concluded that when

   1

, the two estimators and hence their risks coincide.

MLINEX and SE risk functions have been calculated for different values of parameter corresponding to sample sizes and their results are presented in the following tables.

Table 1 : MLINEX and SE risk functions for

  1

^,

   3

^and

n  10

^.

 R

ML

 ^{ ˆ}

B

^{, }  R

ML

 ^{ ˆ}

S

^{, }  R

S

 ^{ ˆ}

B

^{, }  R

S

 ^{ ˆ}

S

^{, } 

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.4270 0.4270 0.4270 0.4270 0.4270 0.4270 0.4270 0.4270 0.4270 0.4270 0.4270 0.4270

0.4694 0.4694 0.4694 0.4694 0.4694 0.4694 0.4694 0.4694 0.4694 0.4694 0.4694 0.4694

0.0584 0.2336 0.5256 0.9344 1.4599 2.1023 2.8615 3.7375 4.7302 5.8398 7.0661 8.4093

0.0417 0.1667 0.3750 0.6667 1.0417 1.5000 2.0417 2.6667 3.3750 4.1667 5.0417 6.0000 Table 2 : MLINEX and SE risk functions for

  2

^,

   2

^and

n  20

^.

 R

ML

 ^{ ˆ}

B

^{, }  R

ML

 ^{ ˆ}

S

^{, }  R

S

 ^{ ˆ}

B

^{, }  R

S

 ^{ ˆ}

S

^{, } 

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.1975 0.1975 0.1975 0.1975 0.1975 0.1975 0.1975 0.1975 0.1975 0.1975 0.1975 0.1975

0.1999 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999

0.0177 0.0708 0.1593 0.2833 0.4426 0.6374 0.8675 1.1331 1.4341 1.7704 2.1422 2.5494

0.0161 0.0643 0.1447 0.2573 0.4020 0.5789 0.7880 1.0292 1.3026 1.6082 1.9459 2.3158 Table 3 : MLINEX and SE risk functions for

  3

^,

   1

and

n  30

.

 R

ML

 ^{ ˆ}

B

^{, }  R

ML

 ^{ ˆ}

S

^{, }  R

S

 ^{ ˆ}

B

^{, }  R

S

 ^{ ˆ}

S

^{, } 

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500

0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500

0.0099 0.0394 0.0887 0.1576 0.2463 0.3547 0.4828 0.6305 0.7980 0.9852 1.1921 1.4187

0.0099 0.0394 0.0887 0.1576 0.2463 0.3547 0.4828 0.6305 0.7980 0.9852 1.1921 1.4187

19

Table 4 : MLINEX and SE risk functions for

  1

^,

  1

^and

n  10

^.

 R

ML

 ^{ ˆ}

B

^{, }  R

ML

 ^{ ˆ}

S

^{, }  R

S

 ^{ ˆ}

B

^{, }  R

S

 ^{ ˆ}

S

^{, } 

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.0556 0.0556 0.0556 0.0556 0.0556 0.0556 0.0556 0.0556 0.0556 0.0556 0.0556 0.0556

0.0613 0.0613 0.0613 0.0613 0.0613 0.0613 0.0613 0.0613 0.0613 0.0613 0.0613 0.0613

0.0313 0.1250 0.2813 0.5000 0.7813 1.1250 1.5313 2.0000 2.5313 3.1250 3.7813 4.5000

0.0417 0.1667 0.3750 0.6667 1.0417 1.5000 2.0417 2.6667 3.3750 4.1667 5.0417 6.0000 Table 5 : MLINEX and SE risk functions for

  2

^,

  2

^and

n  20

^.

 R

ML

 ^{ ˆ}

B

^{, }  R

ML

 ^{ ˆ}

S

^{, }  R

S

 ^{ ˆ}

B

^{, }  R

S

 ^{ ˆ}

S

^{, } 

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.2134 0.2134 0.2134 0.2134 0.2134 0.2134 0.2134 0.2134 0.2134 0.2134 0.2134 0.2134

0.2393 0.2393 0.2393 0.2393 0.2393 0.2393 0.2393 0.2393 0.2393 0.2393 0.2393 0.2393

0.0133 0.0533 0.1200 0.2134 0.3334 0.4801 0.6535 0.8535 1.0802 1.3336 1.6136 1.9203

0.0161 0.0643 0.1447 0.2573 0.4020 0.5789 0.7880 1.0292 1.3026 1.6082 1.9459 2.3158 Table 6: MLINEX and SE risk functions for

  3

^,

  3

and

n  30

.

 R

ML

 ^{ ˆ}

B

^{, }  R

ML

 ^{ ˆ}

S

^{, }  R

S

 ^{ ˆ}

B

^{, }  R

S

 ^{ ˆ}

S

^{, } 

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.4748 0.4748 0.4748 0.4748 0.4748 0.4748 0.4748 0.4748 0.4748 0.4748 0.4748 0.4748

0.5446 0.5446 0.5446 0.5446 0.5446 0.5446 0.5446 0.5446 0.5446 0.5446 0.5446 0.5446

0.0086 0.0345 0.0776 0.1379 0.2155 0.3103 0.4224 0.5517 0.6983 0.8621 1.0431 1.2414

0.0099 0.0394 0.0887 0.1576 0.2463 0.3547 0.4828 0.6305 0.7980 0.9852 1.1921 1.4187

4. Discussion

For any sample size

n

it is evident from the tables that in every case considered except for

   1

, the MLINEX risk function

R

ML

 ^{ ˆ}

B

^{, } 

is uniformly smaller than

R

ML

 ^{ ˆ}

S

^{, } 

. This implies that in case of the MLINEX loss function, the MLINEX estimator

 ^ˆ

_B is better compared to the SE estimator

 ˆ

_S.

It has been seen that when

   1

, the risk of

 ^ˆ

_B with respect to SE loss function is always greater than that of risk of the estimator

 ˆ

_S. Therefore in this case,

 ^ˆ

_S is better compared to the estimator

 ^ˆ

_B ^{for any}

sample size

n

when SE loss function is considered. But if

   1

, it also follows

R

S

 ^{ ˆ}

B

^{, }   R

S

 ^{ ˆ}

S

^{, } 

, implies that

 ^ˆ

_B is an admissible with respect to SE loss function.

It has been further found that when

   1

, the two estimators and hence their risk functions are identical under both quadratic and modified-linear-exponential (MLINEX) loss functions and either estimator

 ^ˆ

_B ^or

 ˆ

_S is an admissible.

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Jeffreys, H. (1961): Theory of Probability, 3^rd ed. Oxford; Clarendon Press.

Madi, M. T. and Raqab, M. Z. (2007): Bayesian prediction of rainfall records using the generalized exponential distribution. Environmetrics, 18, 541-549.

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