Generalized Order Statistics from Generalized Exponential Distributions in Explicit Forms

Generalized Order Statistics from Generalized Exponential

Distributions in Explicit Forms

Hazem I. El Shekh Ahmed Al Quds Open University, Gaza, Palestine

Email: [email protected]

Received January 6,2013; revised February 5, 2013; accepted February 20,2013

Copyright © 2013 Hazem I. El Shekh Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The generalized order statistics which introduced by [1] are studied in the present paper. The Gompertz distribution is widely used to describe the distribution of adult deaths, and some related models used in the economic applications [2]. Previous works concentrated on formulating approximate relationships to characterize it [3-5]. The main aim of this paper is to obtain the distribution of single, two, and all generalized order statistics from Gompertz distribution with some special cases. In addition the conditional distribution of two generalized order statistics from the same distribution is obtained. The Gompertz distribution has a continuous probability density function with location parameter a and

shape parameter b,

 

 

1

e

e ,

bx

a bx

b a







f x where x restricted by the interval



0,



. The nth moment generated function of the Gompertz distributed random variable X is given on the form:

 

e 1 1 ,

a n b

n a

E b

  

   

! n

n E X

b

 where,

 

1

 

ln n e

n

1

d !

s zx

s x x z

n

 

E z





_

is the generalized integro-exponential function [6]. In this paper we shall obtain joint

distribution, distribution of product of two generalized order statistics from the Gompertz distribution, and then derive some useful formulas of these distributions as special cases.

Keywords: Gompertz Distribution; kth Record Model Values; Generalized Order Statistics

1. Introduction

Order statistics appears in many statistical applications and is widely used in statistical modeling and inference. Such models describe random variables arranged in as- cending order of magnitude. In a wide subclass of gener- alized order statistics, representations of marginal, joint and probability density distribution functions are devel- oped. The results are applied to obtain these representa- tions for several expressions for the joint of generalized order statistics from Exponential Pareto distribution [7].

The Gompertz distribution plays an important role in modeling survival times, human mortality and actuarial tables. According to the literature, the Gompertz distri- bution was formulated by Gompertz (1825) to fit mortal- ity tables. On the other hand, generalized order statistics (GOS) have been of interest in the past ten years because they are more flexible in reliability theory, statistical modeling and inference, the generalized order statistics have been introduced as a unified distribution theore-

tical set-up which contains a variety of models of ordered random variables with different interpretations. The sub- ject of order statistics has been further generalized and the concept of generalized order statistics (GOS) is in- troduced and studied by Kamps in a series of papers and books [1,8-10].

1 k sample sizes.

The Generalized Exponential Distribution (GED) with two non-negative parameters  and  is considered to be one of those distributions which have real attention from researchers. It has been studied and introduced by [12].

In the present paper we shall obtain joint distribution and distribution of product of two generalized order sta- tistics from the Gompertz distribution.

2. Generalized Order Statistics (GOS)

Generalized order statistics (GOS) have been of interest in the past ten years because they are more flexible in reliability theory, statistical modeling and inference [12], Uniform generalized order statistics is defined via some joint density function on a cone of Rn. (GOS) based on an arbitrary distribution function F is defined by means of the inverse function of F, as in the following:

Definition (1) Let F x

 

, , ,ΛX



, , ,



denotes an absolutely con- tinuous distribution function with density function f(x), then the sequence of random variables

1: , ,n m k 2: , ,n m k n n m k: , ,

der Statistics (GOS), where (k ≥ 1, m is a real number) [3].

X X is called “n” Generalized Or-

Definition (2) The random variables



1, , ,



, ,

X n m k% Λ X n n m k%

 

are called GOS based on the cumulative distribution function (cdf), F x , if their

joint probability density function (pdf) is given by





 





 

 

1 1

1 ,

1 otherwise

m

j i i

n

x f x

x F

 

 

 

 

 

,n 2,k 0

  

1

n

r j j r

 





 

 

1, , 1

1 1

1 1

1

1 , ,

1

for 0

0,

n n

n n

j i

k

n n

f x x

k F

F x f x

F x



    

  



  

 

    



 

  





Λ Λ

Λ

(1)

with parameters n N ,



1,m2, ,Λmn1



m% m , M m









, such that

0

r k n r Mr

  

r

  r



1, ,Λn1



, 1 n1 _n k

 for all , let

1 1

r j

j

c_  r







 , 2, ,Λ and  



, , ,



.

The cumulative distribution function (cdf) of such gen- eralized order statistics random variable X r n m k

as,

 

 



1

 



1, , 1

1

1 r j j

n r

j j

a r

F X c F X 



 

 



Λ  (2)

where

 

1, 1 r

i i j i j

j

a r

 

 

 

 

 _ 

 



  j r n

0 m

 or1

Now, if  and  it gives the joint pdf of “n” ordinary order statistics X1,n,X2,n, ,Λ Xn n,

1 m

, more- over if  , k1, it gives the joint pdf of the first “n” upper records of the independent identically distrib- uted random variables.

3. The Gompertz Distribution

A random variable X is said to have Gompertz distribu- tion with location parameter a and shape parameter b, has a continuous probability density function (pdf) and for- mulated as,

 

 

e 1

e bx a bx

b f x a









 ,



 

(3)

with a supported domain on . The distribution function (cdf) is:

 

_e 1 1 e bx

a b F x





  (4)

4. Joint Distribution of All Generalized

Order Statistics

4.1. Joint Distribution of All Generalized Order Statistics for Generalized Gompertz

Distribution (GOSGD)

The generalized order statistics were defined by [14] as follows:

Definition (3) Let





1

1 1

, 0, , , n

n

nN k m% m Λm R 

1

0 n

r k n r g rmj

 



be parameters such



1, ,



r Λn . If

  



 for all

that 



, , ,



, 1, , U r n m k% r n

the random variables Λ , possess a

joint density function of the form:



_

_

  



 



1, , , , , , , , 1

1 1

1

1 1

, ,

1 i 1 ,

U n m k U n n m k

n

n n

m k

j i n

j i

f u u

k  u u

  _

 

   

   _  _ 









% Λ %

Λ

1

u u

on the cone 0 1 Λ n

n R

of , then they are called Uniform generalized order statistics. The random variables



_{1, , ,}



1





_{, , ,}





X n m k% F U r n m k%

   

_

_

   





 





 

1, , , , , , , , 1

1 1

1 1

1

, ,

1

1

i

X n m k X n n m k

n

n n

m

j i i

j i

k

n n

f x x

k F x f x

F x f x



    

 

are called generalized order statistics (based on F), which have a joint density function of the form

 

 _{  }  _

 

 

 





% Λ %

Λ

(5)

 

 

1 ₀ 1 ₁

F  X  X  Λ X F

where, 1 2 n

The marginal density function of rth GOS .



, , ,



, 1, ,

X r n m k% r Λn based on F is formulated by





 



, , , 

 

r n:



 

, X r n m k

f x  F x f x

where

     

 

 

 













1 1 : 1 1 ! 0,1 , 1 1 1

ln 1 , 0,1

r r

r

r n m

m m c x x r x

g x m

x   _        _{  }        1 1 , ,if 1 1 , if 1 g x x m x m   _{ }     1 1 r r i i c (6)



 



 

 



 



 







 



 

 

, , , , 1 1 1 , 1

1 ! 1 !

1 where

,

i i j n m k i j

m j

i

i

j m i

j i

m j m i i j

f x x

c

F x

i j i

F x g F x D

D g F x g F x f x f x

        _  _         _{   }   _  _ %

0 _{i j} ,1

and  

 



 





: , r n .

The corresponding marginal distribution of rth GOS based on F is

, , ,

 

X r n m k

F x  F x

where

 





 

1 1 : 0 1 1 ! 0,1 , r r i r n i c x u i u         



1

 

, i m g u      0, m





,X n n, ,0,1

(7)

We discuss some special cases in Corollaries 2.2 and 2.3.

Corollary 4.1 (The joint pdf of all ordinary order statistics for Gompertz Distribution)

In Equation (5), let and then the joint pdf of all ordinary order statistics

1 k



1, ,0,1 ,



X n Λ for Gompertz distribution is











e 1



e e bx i n n a a bx k b b 1

1 1 _{e 1}

1 1

, ,

bx i n

n n _bx

j j i f x x

a a    _{    } 0, m      _{  } 



 



 Λ (8)

Again in Equation (1), let k1 and  then one can directly see that







1 1 1 1 1 n n j j j

n j n n



 

 

     





1



Λ3 2 1  n!. There- fore the joint pdf of all ordinary order statistics for Gompertz Distribution











e 1



e bx

i _n n

a a

bx k

b _a b

1 _{e 1}

1

1

, , ! n ebxi bx n

i

f x x n a

 _{   }



th i _jth



 

 







Λ (9)

We discuss some special cases in Corollaries 2.5 and 2.6.

4.2. Joint Distribution of Two Generalized Order Statistics

The joint pdf of and generalized order statistics,



, , ,



(10)

x x     i j n







1

, 1

r

r j j j

c   k n j m



, where, for 

   







.



X i n m% k and X j n m k



, , ,%



, is given by

Since





1

1

1 1

lim ln 1

1 m m x x m    _{ }           

 





, we shall write

1 1 1 1 m m x g x m    

 for all x

 

0,1 and for all m

 

 

1 lim₁ m

m

g x  _ g x , see [11] with

4.3. Joint Distribution of Two Generalized Order Statistics for Generalized Gompertz

Distribution

In the next theorem 4.1, we will derive thejoint pdf of



, , ,



and



, , ,



X i n m k% X j n m k% for GOSGD.

Theorem 4.1 The joint probability density function of



, , ,



and



, , ,



X i n m k% X j n m k% for Gompertz distribution can be formulated by,











     





 

     1 1 1 2 , , , , 1 1 1 1 1 1 1 1 e e e 2 , e e

1 1 !

1 e 1 where e e e j a a j i b b a i b bx j a bx

a i _b

b

b xi xj a i j b

i j n m k i j

m bx bx j i m bx j i m m

b x x

f x x

c

i j i

D m D a                  _{ } _  _{ }                _{   }                      _{   }     _{ }  _ _   _{ }     %

0 i j ,1

(11)

x x i j n

for      





1

, 1

r

r j j j

c   n j

 , where,    



.  Proof:

 













 





Special cases:

1) The joint density function of two ordinary Order Statistics for Gompertz Distribution

Let and then Equation (11) reduces joint density of two ordinary order statistics

in Gompertz distribution as, 1

k



, , 0,1 ,n



0,

m



, , 0,1



X j n X i











 



 



 

    1 1 1 2

, , ,0,1

1 1 e e e 2 , e 1

1 1 !

where e e e j a j b bx j a bx a i b b

b xi xj a i j b i j n i j

bx j

j i

b x x

f x x

c

i j i

D a                _{ }     _ _        _{ }      _{ }   _{  } _         1 1

e ba i i bx D     _      1

m 



(12)

2) The kth record model values in (GOSGD)

For any and , then Equation (6) di- rectly reduces the kth record model in GOSGD with

kN



ln 1

 

1

g_ x   x

1

, , , n

.

5. Distribution of Single Generalized Order

Statistics

In the following, the concept single generalized order statistics is presented and then going to illustrate the equivalent formulas for the Gompertz distribution.

Definition (4) Further integrating out

1, , r1 r

x Λx x Λ x



, , ,



  from Definition (2), we get the pdf

of

, , ,

r n m k

f % X r n m k as

 





 

, , , 1 ₁ 1 1 ! r

r n m k

r r r m f x c

 



r



 

r

F x g

r

  _

 _  _

 %

F x f x







,  nr m1

(13)

where, and

1

r

r j r j

c   k









 





1

1 1 m

x 

 

 _   _

 

0,1 x 1 1 m g x m

for all for all m, with

 

 

1

lim m x 1

g x g x





 1 1 1

_{ }

1 1 m f x    1 r  , see [14].

Lemma 5.1 The pdf of the minimum generalized order

statistic is

 







 

1, , ,

1 1 1

n m k

k n

f x

k n m F x  

_     _{ } 

%

(14)

Proof:

Using Equation (13), let  , then







1 1

c   k n1 m1 , we get Equation (14) and

that completes the proof.

Lemma 5.2 The pdf of the maximum generalized or-

der statistic is

 

 





 

, , , 1 1 1

1 1 2 1

1 !

1

1 _{1 1}

1 n n m k

k n n m n n f x

k n m k n m k

n F x

F x f x

m                       _{ }  _{ } _{ }   _    % Λ

rn

 





(15)

Proof:

Using Definition (4), let , then

 





1

1

1 1 ,

1

m

m F xn F xn n k

m 



 

g  _   _ 





















 

1

1

1 1 2 1

n

n j

c k n j m

k n m k n m k

     _     _{ }    _



Λ ,

we get Equation (15) and that completes the proof.

Distribution of Single Generalized Order Statistics for Gompertz Distribution

Lemma 5.3 Using the pdf and cdf given in Equations (2)

and (3) in Equation (13), and collecting terms we get

 

















, , , 1 1 1

e 1 e 1

e 1

1

e 1 e

1 ! 1

e

r

bxr bx

bxr r n m k

r m

a a

r b b

a bxr b f x c r m a               _   _    _   _{  }   _    _{ }   _  % 1 r (16)

Corollary 5.1 The pdf of the minimum generalized order statistic for Gompertz Distribution

In Equation (16), let  ,







1 1 1 1

c   k n m , and collecting terms then

 

 

the pdf of the minimum generalized order statistic for GompertzDistribution is

 











  





1, , ,

1 1 1

e 1 e 1

1 1

e bxi e i bxi

n m k

k n m

a a

bx

b b

f x

k n m

a         _    _        % 1 k (17)

Corollary 5.2 The pdf of the minimum ordinary statistics for Gompertz Distribution

In Equation (17), let  and m0 then,

 













1

e 1 e 1

1, ,0,1 1 1 e e

bxi bxi

i n a a bx b b n

f x n a



   

 

 _  _{  }

 

Lemma 5.3 Using the pdf and cdf given in Equations

 







_



_





 













, , ,

1 1

e 1 e 1

1 1 2 1

1 !

1 _{1 e e}

1

bxn _n bxn

n n m k

n m

a a

bx

b b

k n m k n m k

f x

n

a m

 

   

     

  

  



  _{ } 

  

_{  } _ _{ } 



 

 

 

%

Λ e 1 1

e bxn k a b

  

  



 _ _ 

 

1

k m0

(18)

6. Conditional Distribution of Generalized

Order Statistics

Corollary 5.4 The pdf of the maximum generalized order statistic for Gompertz Distribution

In Equation (18), let and , then the pdf of the maximum ordinary order statistic for Gompertz Distribution

 









1

e 1 e 1

e

bxn bxn

n n

a a

bx

b _a b



   

  

In this section some previous literature of the conditional distribution of generalized order statistics is presented and then derived these results for Gompertz distribution.

, ,0,1 1 e

n n

f x n_ _ __

 

 







(19)

where,







 

















1 1

k n m k n

   2 1

1 !

1 3 2 1 !

1 ! 1 !

m k

n

n n n

n

n n

 

   

      

  

 

Λ

Λ

, ,

Theorem 5.1 (See [15])

Let X1Λ Xn be a random sample from a continuous population with cdf F

 

x and pdf f x

 

. Let



1, , ,

 

, 2, , ,



, ,



, , ,



X n m k% X n m k% Λ X n n m k% denote the

generalized order statistics obtained from this sample. Then the conditional distribution of



 



X s n m k X r n m k, , ,% , , ,% xforrs is



 











 

 



 





 



 

 

1 1

1

, , , , , ,

1 1

1 ! ₁

s

r

s r m

m m

s r

h X s n m k X r n m k

F x F y g F y g F x f y

c

c s r _{F x}





  



 

  

   

     

 

  _{  }

% %

 

 

1 ₀ 1 ₁

F   x y F

1

r

r j j

c

(20)

 

 

1 lim₁ m

m

Notes 







, j k



n j



m1



and

 





m1

x 

 _ 

1 1 1 1

g x

m

 _  _

 x

 

0,1

m

m

g

all with

for all and for

x g x

  _

1

k m0





.

Corollary 5.5 (Conditional distribution of an ordinary

order statistics)

Using Theorem (5.1), if and , then









 

 



 





 



 

 













1

0 1

0 0

1

1 1

, ,0,1 ,

1 1 !

, ,0,1 , ,0,1

s

r

s r

F _s

, ,0,1

r

x F y g F y g F x f y c

X r n

F x c s r

x r n y s n





  



 

  

   

     



  

 

 

  

 





 





 

1

0 r

n r

g F y

g F x F x

 _

  

 









 

_{  }

h X s n



 











 



 





 















 

1 , 1 ,

1 1

r s n s

n r n r

F y

    

   

_

_{ }

_

_{ }

 

1

, ! 1

s r

n r

F x f y

F x

 



  

 



 

 

, ,0,1 , ,0,1

! 1

! 1

n s h X s n X r n

n r F y F y

n s s r



 _  _



  

(21)

0

! ! n n s

 



_

_

Conditional Distribution of Generalized Order Statistics for Gompertz Distribution

1 n2Λ1 n s s

c n n , and

Lemma 5.5 Using Equation (20) The conditional pdf

of



1



2





!

! n

n n r

n r

   



Λ1

r



 



































1 1

e 1

e

by m

a by

b a



 

1

k m0







1 1

e 1 e 1 e 1 e 1

1 e 1

, , , , , ,

1

e e e e

1

1 !

e

s

bx by by bx

r bx

s r

m m

a a a a

b b b b

s

a r

b h X s n m k X r n m k

m c

c s r





 

 

       

  

  

    _     _

    

    _{      }



    _{ }    _

 

  _{ }

 

 

% %

(22)

Corollary 5.6

In Equation (22), let and , and collecting terms then the conditional pdf oftwo ordinary order sta- tistics, X s n, ,0,1



, ,0,1





X r n



x





, for Gompertzdistri- bution is











 























1 1

e 1

, ,0,1 !

! 1 !

e e e e

e

s

by by bx by

r bx

s r

a a a a

by

b b b b

a b h X s n X r

n r n s s r

a





  

       

  



 

  

   



   

   



 

 

 

e 1 e 1 e 1

1 e 1

, ,0,1n

(23)

where,









 





! ,

! 1 !

1

s n r

c

s s r

n s

     1 !

1 ,

r

r s

c s r n

r n

 



  

    

which is the well known conditional distribution of two ordinary order statistics X s n



, ,0,1

 

X r n, ,0,1



for LGD.

7. Conclusion and Future Research

In this paper, we have derived the joint pdfs of general- ized order statistics for Generalized and Linear Exponen- tial distributions in explicit forms. In addition, the pdf of the conditional distribution of generalized order statistics from those distributions is given. Furthermore, some spe- cial cases have been discussed.

Many opportunities of future research are available. The plan for the future research on generalized order Statistics from Generalized and Linear Exponential dis- tributions can be split into two main areas. Estimation and hypothesis testing of Generalized Exponential pa- rameters based on generalized order statistics.

8. Acknowledgements

The author would like to thank the lecture Shaikh Ahmed R. for her efforts in this research and to Mr. Fayyad M. for his technical aid.

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