Computing the Moments of Order Statistics from Independent Nonidentically Distributed Exponentiated Frechet Variables

The moments of order statistics o.s. arising from independent nonidentically distributed inid three parameter Exponentiated Frechet EF random variables r.v.’s. were computed using a theorem of Barakat and Abdelkader 2003 . Two methods of integration were used to find the moments. Graphical representation of the probability density function p.d.f. and the cumulative distribution function c.d.f. of the rth o.s. arising from inid r.v.’s. from this distribution. Calculations of the mean of the largest o.s. from a sample of size 2 were given for both inid and independent identically distributed iid r.v.’s.


Introduction
Nadarajah and Kotz 1 introduced a new lifetime model named the Exponentiated Frechet distribution EF.It is a generalization of the standard Frechet distribution known as the extreme value distribution of type II .The EF distribution is referred to in the literature as the inverse of exponentiated Weibull distribution.The cumulative distribution function c.d.f. of the EF can be written as , x > 0, σ > 0, λ > 0, α > 0, 1.1 where α and λ are the shape parameters and σ is the scale parameter, respectively.They provided a comprehensive treatment of the mathematical properties of this new distribution such as the derivation of the analytical shapes of the corresponding probability density function, the hazard rate function and provided graphical illustrations.They also

Nonidentical Order Statistics from Exponentiated Frechet Distribution
The subject on nonidentical order statistics is discussed widely in the literature in David and Nagaraja 4 .Vaughan and Venables 5 denoted the joint p.d.f. and marginal p.d.f. of order statistics of inid random variables by means of the permanent.Let X 1 , X 2 , . . ., X n be independent random variables having cumulative distribution functions F 1 x , F 2 x , . . ., F n x and probability density functions f 1 x , f 2 x , . . ., f n x , respectively.Let X 1:n ≤ X 2:n ≤ • • • ≤ X n:n denote the order statistics obtained by arranging the n X , i s in increasing order of magnitude.Then the p.d.f. and the c.d.f. of the rth order statistic X r:n 1 ≤ r ≤ n can be written as where p denotes the summation over all n! permutations i 1 , i 2 , . . ., i n of 1, 2, . . .n .Bapat and Beg 6 put it in the form of the permanent as

2.2
The p.d.f. and the c.d.f. of the rth inid o.s. of EF distribution are displayed in Figures 1 and 2 for some selected values of the shape parameters λ and α i , i 1, 2, 3 and for the scale parameter σ 1, when the sample size n 3, r 1, 2, 3.Where p j is all permutations of i 1 , i 2 , . . ., i n for 1, . . ., n which satisfy And using the permanent, we have For EF distribution, we have 2.4   In this section, the theorem which was established by Barakat and Abdelkader 3 will be stated without proof.Then the theorem is used to get recurrence relation for the single moments of inid o.s.arising from EF distribution.Theorem 3.1.Let X 1 , X 2 , . . ., X n be independent nonidentically distributed r.v.'s.The kth moment of the rth o.s.μ k r:n , for 1 ≤ r ≤ n and k 1, 2, . . . is given by

The
where The following theorem gives an explicit expression for I j k when X 1 , X 2 , . . ., X n are inid EF r.v.'s.Two cases were considered.

Case 1 (When The Shape Parameter α of EF Distribution Is an Integer)
where or where Proof.On applying Theorem 3.1 and using 2.4 , we get 1 − e − σ/x λ α i t dx.

3.7
This integral converges if j t 1 α i t > k/λ, and we used two methods of integration.

The First Method
The first method is to find this integral by expanding the expression 1 z a .When the exponent a is an integer, we will use the series Using the Gamma notation and considering b 1, this series can be written as 3.9

Journal of Probability and Statistics
Then the bracket in 3.7 can be written as

3.11
Making some arrangements, we get 3.3 .

The Second Method of Integration is Given by Using the Transformation
y e − σ/x λ , 3.12 in 3.7

3.16
Table 1: I j k using 3.3 when n 3.
Table 2: I j k using 3.5 when n 3.

3.17
Substituting 3.17 in 3.15 and making some arrangements, we get 3.5 .

3.19
The iid case can be deduced from Theorem 3.2.The result will be stated in the next corollary.
Corollary 3.4.For the case of a sample of n iid r.v.'s having EF distribution, the I j k in Theorem 3.2 simply reduces to where S j jα.

Numerical Applications
The following examples are computed when k 1.

Case 2 (When the Shape Parameter α of EF Is Noninteger)
When α is noninteger, the expansion of 1 z a is then written as

3.28
Equations 3.3 and 3.5 become where where

3.33
The iid case can be deduced from 3.29 .The result will be stated in the next corollary.
Corollary 3.9.For a sample of iid r.v. ,s having EF distribution, the I j k in 3.29 simply reduces to Example 3.11.Let n 2 and α 1.5, 2.5, 3.5, 4.5.Table 5 shows the results of calculations.
Remark 3.12.Calculations in Example 3.11 were done following the method of Abdelkader 12 .The upper limit of the sum ∞ is taken up to α , where . is the usual greatest integer function.
Remark 3.13.All figures and tables in this paper had been accomplished by Mathematica 7.0.

Moments of the rth o.s. Arising from Independent Nonidentically Distributed Exponentiated Frechet Random Variables
to derive the moments of inid order statistics for several distributions.