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

Authors: A. A. Jamjoom; Z. A. Al-Saiary
Journal: Journal of Probability and Statistics
 (2012)

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## Abstract

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 therth 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.


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## Body

## 1. 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(1.1)F(x)=1-[1-e-(σx)λ]α,x>0,σ>0,λ>0,α>0,
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 calculated expressions for thenth moment, the asymptotic distribution of the extreme order statistics, investigated the variation of the skewness and kurtosis, and discussed estimation by the method of maximum likelihood.This distribution was extensively studied by Badr [2] in a P.H.d. dessertation from several statistical points of view such as statistical properties, relation between the EF and several other distributions, statistical inferences, order statistics, record values, and associated inference.The subject of nonidentical order statistics o.s. for EF is not discussed in the literature yet for EF distribution, which was the motivation behind this paper.Mathematical and graphical representation of the probability density function p.d.f. and the c.d.f. of therth o.s. arising from inid EF distribution are given in Section 2.Computation of moments of therth o.s. of inid r.v.’s. arising from inid EF using Barakat and Abdelkader [3] technique is presented in Section 3.This technique requires that the c.d.f. of the distribution can be written in the formF(x)=1-λ(x), which is satisfied in this distribution. This technique is referred to as Barakat Abdelkader technique (BAT).


## 2. 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.LetX1,X2,…,Xn  be independent random variables having cumulative distribution functions F1(x),F2(x),…,Fn(x) and probability density functions f1(x),f2(x),…,fn(x), respectively. Let X1:n≤X2:n≤⋯≤Xn:n  denote the order statistics obtained by arranging the nXi,s in increasing order of magnitude. Then the p.d.f. and the c.d.f. of the rth order statistic Xr:n(1≤r≤n) can be written as(2.1)fr:n(x)=1(r-1)!(n-r)!∑p∏a=1r-1Fia(x)fir(x)∏c=r+1n{1-Fic(x)},
where ∑p  denotes the summation over all n! permutations (i1,i2,…,in) of (1,2,…n). Bapat and Beg [6] put it in the form of the permanent as(2.2)fr:n(x)=1(r-1)!(n-r)!per[F(x)︸r-1f(x)︸1{1-F(x)}︸n-r],F(r)(x)=∑j=rn∑pj∏a=1jFia(x)∏a=j+1n[1-Fia(x)].
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 αi, i=1,2,3 and for the scale parameter σ=1, when the sample size n=3, r=1,2,3.Figure 1
Graph of p.d.f. of (inid) o.s.  from Ef distribution for selected values ofα1=0.03; α2=0.05; α3=0.07; σ=1, n=3, r=1,2,3, respectively.Figure 2
Graph of c.d.f. of inid o.s. from Ef distribution for selected values ofα1=0.03; α2=0.05; α3=0.07; σ=1, n=3, r=1,2,3,  respectively.Where∑pj  is all permutations of (i1,i2,…,in) for (1,…,n)which satisfy i1<i2<⋯<ij  and ij+1<ij+2<⋯<in. And using the permanent, we have(2.3)F(r)(x)=∑i=rn1i!(n-i)!per[F1(x)1-F1(x)⋮Fn(x)︸i1-Fn(x)︸n-i],-∞<x<∞.
For EF distribution, we have(2.4)Fi(x)=1-[1-e-(σ/x)λ]αi,x>0,σ>0,λ>0,αi>0,fi(x)=αiλσλ+1x-(λ+1)[1-e-(σ/x)λ]αi-1e-(σ/x)λ,x>0,σ>0,λ>0,αi>0.


## 3. The Moments of therth o.s. Arising from Independent Nonidentically Distributed Exponentiated Frechet Random Variables

Three techniques have been established in the literature to compute moments of o.s. of inid r.v.’s. see Balakrishnan [7], Barakat and Abdelkader [3], and Jamjoom and Al-Saiary [8]. Applications of the previous two methods are also found in the literature for several continuous distributions. The paper [7] established the first technique which was later referred to as differential equation technique (DET) and used it to derive recurrence relations for single and product moments of inid order statistics from the Exponential and right truncated distributions. Childs and Balakrishnan [9] applied (DET) to derive the moments of inid order statistics for logistic random variables. Mohie Elidin et al. [10] applied this method to derive the moments of inid order statistics for several distributions.Barakat and Abdelkader [11] established the second technique and applied it to Weibull distribution. They generalized it in Barakat and Abdelkader [3] and applied it to several continuous distributions such as Erlang, Positive Exponential, Pareto, and Laplace distribution. This method was also used to compute the moments of inid o.s. of Gamma distribution in Abdelkader [12], Burr type XII distribution in Jamjoom [13], Beta distribution in Abdelkader [14]. Later, in Jamjoom and Al-Saiary, [15] was referred to as (BAT) and had been used to compute the moments of inid o.s. of Beta three-parameter type I distribution.The third technique, established by Jamjoom and Al-Saiary [8], is the moment generating function technique. It depends mainly on BAT. It is referred to as (M.G.F BAT) and it was used by the same authors to compute the moments of inid o.s. for Burr type II distribution, Exponential distribution and Erlang truncated Exponential distribution.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.
LetX1,X2,…,Xnbe independent nonidentically distributed r.v.’s. The kth moment of the rth o.s. μr:n(k), for 1≤r≤n and k=1,2,… is given by
(3.1)μr:n(k)=∑j=n-r+1n(-1)j-(n-r+1)(j-1n-r)Ij(k),
where
(3.2)Ij(k)=∑1≤i1<i2<⋯<ij≤n…∑k∫0∞xk-1∏t=1jGit(x)dx,j=1,2,…,n,Git(x)=1-Fit(x), with (i1,i2,…,in) are permutations of (1,2,…,n) for which i1≤i2<⋯<in.
The following theorem gives an explicit expression forIj(k)when X1,X2,…,Xnare inid EF r.v.’s. Two cases were considered.


### 3.1. Case  1 (When The Shape Parameterα of EF Distribution Is an Integer)

Theorem 3.2.
For1≤r≤n,k=1,2,…,
(3.3)Ij(k)=kσkΓ(-k/λ)λ∑1≤i1<i2<⋯∑<ij≤n∑m1=0αi1⋯∑mj=0αijξij(∑t=1jmt)k/λ,
where
(3.4)ξij=∏t=1j(-1)mtΓ(αit+1)mt!Γ(αit-mt+1),
or
(3.5)Ij(k)=kσkΓ(-k/λ)λ∑1≤i1<i2<⋯∑<ij≤n∑m=0Sij(Sijm)(-1)Sij-m(Sij-m)k/λ,
where
(3.6)Sij=∑t=1jαit.Proof.
On applying Theorem3.1 and using (2.4), we get
(3.7)Ij(k)=∑1≤i1<i2<⋯∑<ij≤nk∫0∞xk-1∏t=1j[1-e-(σ/x)λ]αitdx.
This integral converges if  ∑t=1jαit>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
(3.8)(b+z)a=∑m=0a(am)ba-mzm,
see Abramowitz and Stegun [16] page 10.
Using the Gamma notation and consideringb=1, this series can be written as
(3.9)(1+z)a=∑m=0aΓ(a+1)Γ(a-m+1)zmm!.
Then the bracket in (3.7) can be written as
(3.10)[1-e-(σ/x)λ]αit=∑m=0αitΓ(αit+1)Γ(αit-m+1)(-1)me-m(σ/x)λm!∴Ij(k)=∑1≤i1<i2<⋯∑<ij≤nk∫0∞xk-1∏t=1j∑m=0αitΓ(αit+1)Γ(αit-m+1)(-1)me-m(σ/x)λm!dx=∑1≤i1<i2<⋯∑<ij≤nk∫0∞xk-1(×(-1)∑t=1jmte-∑t=1jmt(σ/x)λ∏t=1jmt!∑m1=0αi1Γ(αi1+1)Γ(αi1-m1+1)(-1)m1e-m1(σ/x)λm1!⋯∑mj=0αijΓ(αit+1)Γ(αij-mj+1)×(-1)mje-mj(σ/x)λmj!×(-1)∑t=1jmte-∑t=1jmt(σ/x)λ∏t=1jmt!)dx∴Ij(k)=∑1≤i1<i2<⋯∑<ij≤nk∫0∞xk-1(×(-1)∑t=1jmte-∑t=1jmt(σ/x)λ∏t=1jmt!∑m1=0αi1⋯∑mj=0αijΓ(αi1+1)Γ(αi1-m1+1)⋯Γ(αij+1)Γ(αij-mj+1)×(-1)∑t=1jmte-∑t=1jmt(σ/x)λ∏t=1jmt!)dx=∑1≤i1<i2<⋯∑<ij≤nk∑m1=0αi1⋯∑mj=0αij(∏t=1jΓ(αit+1)Γ(αit-mt+1)(-1)mtmt!×∫0∞xk-1e-∑t=1jmt(σ/x)λdx)=∑1≤i1<i2<⋯∑<ij≤nk∑m1=0αi1⋯∑mj=0αij(∏t=1j(-1)mtmt!Γ(αit+1)Γ(αit-mt+1)×∫0∞xk-1e-∑t=1jmt(σ/x)λdx).
Substituting
(3.11)y=∑t=1jmt(σx)λ∴x=σ(∑t=1jmt)1/λy-1/λdx=σλ(∑t=1jmt)1/λy-(1/λ)-1dy∴∫0∞xk-1e-∑t=1jmt(σ/x)λdx=σk(∑t=1jmt)k/λ∫0∞y-(k/λ)-1e-ydy=σkλ(∑t=1jmt)k/λΓ(-kλ);λ>0,k=1,2,…,kλ≠0,1,2,….
Making some arrangements, we get (3.3).The Second Method of Integration is Given by Using the Transformation
(3.12)y=e-(σ/x)λ,
in (3.7)
(3.13)∴x=σ(ln⁡1y)-1/λ,
Then
(3.14)dx=-σλ(ln⁡1y)-(1/λ)-1y-1dy∴∫0∞xk-1∏t=1j[1-e-(σ/x)λ]αitdx=∫0∞xk-1[1-e-(σ/x)λ]Sijdx,
where
(3.15)Sij=∑t=1jαit=σkλ∫01(ln⁡1y)-(k/λ)-1[1-y]Sijy-1dy=σkλ∑m=0Sij(Sijm)(-1)Sij-m∫01(ln⁡1y)-(k/λ)-1ySij-m-1dy.
Using
(3.16)∫01(ln⁡1y)θ-1yν-1dy=1νθΓ(θ),
see (4.272.6) in Gradshteyn and Ryzhik [17]
(3.17)∴∫01(ln⁡1y)-(k/λ)-1ySij-m-1dy=1(Sij-m)-k/λΓ(-kλ);k≠0,λ,2λ,3λ,….
Substituting (3.17) in (3.15) and making some arrangements, we get (3.5).Tables1 and 2 give the values of  Ij in (3.3) and (3.5), for  j=1,2,3 and sample size n=3.Table 1
Ij(k) using (3.3) when n=3.
jIj(k)1I1(k)=kσkΓ(-k/λ)λ(∑m1=0α1(-1)m1Γ(α1+1)m1!Γ(α1-m1+1)(m1)k/λ+∑m1=0α2(-1)m1Γ(α2+1)m1!Γ(α2-m1+1)(m1)k/λ+∑m1=0α3(-1)m1Γ(α3+1)m1!Γ(α3-m1+1)(m1)k/λ)2I2(k)=kσkΓ(-k/λ)λ(∑m1=0α1∑m2=0α2(-1)m1+m2Γ(α1+1)Γ(α2+1)m1!m2!Γ(α1-m1+1)Γ(α2-m2+1)(m1+m2)k/λ+∑m1=0α1∑m2=0α3(-1)m1+m2Γ(α1+1)Γ(α3+1)m1!m2!Γ(α1-m1+1)Γ(α3-m2+1)(m1+m2)k/λ+∑m1=0α2∑m2=0α3(-1)m1+m2Γ(α2+1)Γ(α3+1)m1!m2!Γ(α2-m1+1)Γ(α3-m2+1)×(m1+m2)k/λkσkΓ(-k/λ)λ)3I3(k)=kσkΓ(-k/λ)λ(∑m1=0α1∑m2=0α2∑m3=0α3(-1)m1+m2+m3Γ(α1+1)Γ(α2+1)Γ(α3+1)m1!m2!m3!Γ(α1-m1+1)Γ(α2-m2+1)Γ(α2-m2+1)×(m1+m2+m3)k/λkσkΓλ)Table 2
Ij(k) using (3.5) when n=3.
jIj(k)1I1(k)=kσkΓ(-(k/λ))λ∑i=1n∑m=0αi(αim)(-1)αi-m(αi-m)k/λ2I2(k)=kσkΓ(-k/λ)λ(kσkΓλ∑m=0α1+α2(α1+α2m)(-1)α1+α2-m(α1+α2-m)k/λ+∑m=0α1+α3(α1+α3m)(-1)α1+α3-m(α1+α3-m)k/λ+∑m=0α2+α3(α2+α3m)(-1)α2+α3-m(α2+α3-m)k/λkσkΓλ)3kσkΓ(-k/λ)λ(kσkΓλ∑m=0α1+α2+α3(α1+α2+α3m)(-1)α1+α2+α3-m(α1+α2+α3-m)k/λ(α1+α2+α3m))Corollary 3.3.
For inid EF r.v.’s, when j  = 1, (3.3) becomes
(3.18)I1(k)=kσkΓ(-k/λ)λ∑i=1n∑r=1αi(-1)rαi(αi-1)⋯(αi-(r-1))rk/λr!.Proof.
One has(3.19)I1(k)=kσkΓ(-k/λ)λ∑i=1n(∑m1=0αiξimtk/λ)=kσkΓ(-k/λ)λ∑i=1n(∑m1=0αi(-1)m1Γ(αi+1)m1!Γ(αi-m1+1)mtk/λ)=kσkΓ(-k/λ)λ×∑i=1n(0-Γ(αi+1)Γ(αi)+Γ(αi+1)2Γ(αi-1)2k/λ-Γ(αi+1)3!Γ(αis-2)3k/λ+Γ(αi+1)4!Γ(αi-2)4k/λ+⋯+(-1)αiΓ(αi+1)αi!Γ(1)αik/λ)=kσkΓ(-k/λ)λ×∑i=1n(-αi+αi(αi-1)22k/λ-αi(αi-1)(αi-2)3!3k/λ+αi(αi-1)(αi-2)(αi-3)4!4k/λ+⋯+(-1)αiαik/λαi(αi-1)2)=kσkΓ(-k/λ)λ∑i=1n(∑r=1α(-1)r+αi(αi-1)+⋯+(αi-(r-1)rk/λr!)).
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, theIj(k)in Theorem 3.2 simply reduces to
(3.20)Ij(k)=kσkΓ(-k/λ)λ(nj)∑m1=0α⋯∑mj=0αξij(∑t=1jmt)k/λ,
where
(3.21)ξj=∏t=1j(-1)mtΓ(α+1)mt!Γ(α-mt+1),
or
(3.22)Ij(k)=kσkΓ(-k/λ)λ(nj)∑m=0Sj(Sjm)(-1)Sj-m(Sj-m)k/λ,
where
(3.23)Sj=jα.Corollary 3.5.
For iid EF r.v.’s, when j  = 1, (3.18) becomes
(3.24)I1(k)=knσkΓ(-k/λ)λ∑r=1α(-1)rα(α-1)⋅⋅⋅(α-(r-1))rk/λr!.Numerical Applications
The following examples are computed whenk=1.Example 3.6.
Letn=2 and α=2,3,4, and 5. Table 3 shows the results of the calculations.
For example, whenα=3, σ=1, λ=2,
(3.25)μ2:2=I1-I2,I1=2Γ(-1/2)2∑r=13(-1)rα(α-1)⋅⋅⋅(α-(r-1))rk/λr!=π(-α+2α(α-1)2-2α(α-1)(α-2)6)=1.73491I2=Γ(-1/2)2(22)∑m1=03∑m2=03∏t=12(-1)tΓ(α+1)mj!Γ(α-mj+1)(∑t=12mt)1/2=0.699642μ2:2=1.73491-0.699642=1.03527.Table 3
The momentsμ2:2 arising from iid Exponentiated Frechet r.v.’s. using (3.3) or (3.5).
α2345μ2:21.29151.035270.9168920.84626Example 3.7.
Settingn=2, σ=1, λ=2, and α1=1(1)5, α2=1(1)5 in Theorems 3.1 and 3.2, we get the results of the calculations in Table 4.
For example, whenα1=2,α2=3(3.26)μ2:2=I1-I2,I1=Γ(-1/2)λ∑i=12∑r=1∞(-1)rαi(αi-1)⋅⋅⋅(αi-(r-1))rk/λr!=Γ(-1/2)λ(∑r=1α1(-1)rα1(α1-1)⋯(α1-(r-1))rk/λr!+∑r=1α2(-1)rα2(α2-1)⋯(α2-(r-1))rk/λr!)=1.90574I2=Γ(-1/2)2∑m1=02∑m2=03(-1)m1+m2Γ(α1+1)Γ(α2+1)mt!m!Γ(α1-m1+1)Γ(α2-m2+1)(m1+m2)1/2=0.734577μ2:2=1.90574-0.734577=s1.17117.Table 4
The momentsμ2:2arising from inid Exponentiated Frechet r.v.’s. using (3.3) or (3.5).
α2α11234512.506631.943281.854851.822941.8073921.943281.29151.171161.12371.0992531.854851.171161.035270.9789050.94880841.822941.12370.9789050.9168920.88295351.807391.099250.9488080.8829530.84626


### 3.2. Case  2 (When the Shape Parameterα of EF Is Noninteger)

Whenα  is noninteger, the expansion of (1+z)a  is then written as(3.27)(1+z)a=∑m=0∞(am)zm,-1<z<1,see Abramowitz and Stegun [16] page 14. Using gamma notation, (3.27) can be written as(3.28)(1+z)a=∑m=0∞Γ(a+1)Γ(a-m+1)zmm!,-1<z<1.Equations (3.3) and (3.5) become(3.29)Ij(k)=kσkΓ(-k/λ)λ∑1≤i1<i2<⋯∑<ij≤n∑m1=0∞⋯∑mj=0∞ξij(∑t=1jmt)k/λ,
where
(3.30)ξij=∏t=1j(-1)mtΓ(αit+1)mt!Γ(αit-mt+1),
or(3.31)Ij(k)=kσkΓ(-k/λ)λ∑1≤i1<i2<⋯∑<ij≤n∑m=0∞(Sijm)(-1)Sij-m(Sij-m)k/λ,
where
(3.32)Sij=∑t=1jαit.Corollary 3.8.
For inid EF r.v.’s, when j  = 1, (3.29) becomes
(3.33)I1(k)=kσkΓ(-k/λ)λ∑i=1n∑r=1∞(-1)rαi(αi-1)⋅⋅⋅(αi-(r-1))rk/λr!.
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 Ij(k)in (3.29) simply reduces to
(3.34)Ij(k)=kσkΓ(-k/λ)λ(nj)∑m1=0∞⋯∑mj=0∞ξij(∑t=1jmt)k/λ,
where
(3.35)ξj=∏t=1j(-1)mtΓ(α+1)mt!Γ(α-mt+1),
or
(3.36)Ij(k)=kσkΓ(-k/λ)λ(nj)∑m=0∞(Sjm)(-1)Sj-m(Sj-m)k/λ,
where
(3.37)Sj=jα.Corollary 3.10.
For iid EF r.v.’s, when j  = 1, (3.33) becomes
(3.38)I1(k)knσkΓ(-k/λ)λ∑r=1∞(-1)rα(α-1)⋅⋅⋅(α-(r-1))rk/λr!.Example 3.11.
Letn=2 and α=1.5,2.5,3.5,4.5. Table 5 shows the results of calculations.Table 5
The momentsμ2:2arising from IID Exponentiated Frechet r.v.’s. using (3.29) or (3.31).
α1.52.53.54.5μ2:22.684690.6663081.302360.637316Remark 3.12.
Calculations in Example3.11 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.


## 3.1. Case  1 (When The Shape Parameterα of EF Distribution Is an Integer)

Theorem 3.2.
For1≤r≤n,k=1,2,…,
(3.3)Ij(k)=kσkΓ(-k/λ)λ∑1≤i1<i2<⋯∑<ij≤n∑m1=0αi1⋯∑mj=0αijξij(∑t=1jmt)k/λ,
where
(3.4)ξij=∏t=1j(-1)mtΓ(αit+1)mt!Γ(αit-mt+1),
or
(3.5)Ij(k)=kσkΓ(-k/λ)λ∑1≤i1<i2<⋯∑<ij≤n∑m=0Sij(Sijm)(-1)Sij-m(Sij-m)k/λ,
where
(3.6)Sij=∑t=1jαit.Proof.
On applying Theorem3.1 and using (2.4), we get
(3.7)Ij(k)=∑1≤i1<i2<⋯∑<ij≤nk∫0∞xk-1∏t=1j[1-e-(σ/x)λ]αitdx.
This integral converges if  ∑t=1jαit>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
(3.8)(b+z)a=∑m=0a(am)ba-mzm,
see Abramowitz and Stegun [16] page 10.
Using the Gamma notation and consideringb=1, this series can be written as
(3.9)(1+z)a=∑m=0aΓ(a+1)Γ(a-m+1)zmm!.
Then the bracket in (3.7) can be written as
(3.10)[1-e-(σ/x)λ]αit=∑m=0αitΓ(αit+1)Γ(αit-m+1)(-1)me-m(σ/x)λm!∴Ij(k)=∑1≤i1<i2<⋯∑<ij≤nk∫0∞xk-1∏t=1j∑m=0αitΓ(αit+1)Γ(αit-m+1)(-1)me-m(σ/x)λm!dx=∑1≤i1<i2<⋯∑<ij≤nk∫0∞xk-1(×(-1)∑t=1jmte-∑t=1jmt(σ/x)λ∏t=1jmt!∑m1=0αi1Γ(αi1+1)Γ(αi1-m1+1)(-1)m1e-m1(σ/x)λm1!⋯∑mj=0αijΓ(αit+1)Γ(αij-mj+1)×(-1)mje-mj(σ/x)λmj!×(-1)∑t=1jmte-∑t=1jmt(σ/x)λ∏t=1jmt!)dx∴Ij(k)=∑1≤i1<i2<⋯∑<ij≤nk∫0∞xk-1(×(-1)∑t=1jmte-∑t=1jmt(σ/x)λ∏t=1jmt!∑m1=0αi1⋯∑mj=0αijΓ(αi1+1)Γ(αi1-m1+1)⋯Γ(αij+1)Γ(αij-mj+1)×(-1)∑t=1jmte-∑t=1jmt(σ/x)λ∏t=1jmt!)dx=∑1≤i1<i2<⋯∑<ij≤nk∑m1=0αi1⋯∑mj=0αij(∏t=1jΓ(αit+1)Γ(αit-mt+1)(-1)mtmt!×∫0∞xk-1e-∑t=1jmt(σ/x)λdx)=∑1≤i1<i2<⋯∑<ij≤nk∑m1=0αi1⋯∑mj=0αij(∏t=1j(-1)mtmt!Γ(αit+1)Γ(αit-mt+1)×∫0∞xk-1e-∑t=1jmt(σ/x)λdx).
Substituting
(3.11)y=∑t=1jmt(σx)λ∴x=σ(∑t=1jmt)1/λy-1/λdx=σλ(∑t=1jmt)1/λy-(1/λ)-1dy∴∫0∞xk-1e-∑t=1jmt(σ/x)λdx=σk(∑t=1jmt)k/λ∫0∞y-(k/λ)-1e-ydy=σkλ(∑t=1jmt)k/λΓ(-kλ);λ>0,k=1,2,…,kλ≠0,1,2,….
Making some arrangements, we get (3.3).The Second Method of Integration is Given by Using the Transformation
(3.12)y=e-(σ/x)λ,
in (3.7)
(3.13)∴x=σ(ln⁡1y)-1/λ,
Then
(3.14)dx=-σλ(ln⁡1y)-(1/λ)-1y-1dy∴∫0∞xk-1∏t=1j[1-e-(σ/x)λ]αitdx=∫0∞xk-1[1-e-(σ/x)λ]Sijdx,
where
(3.15)Sij=∑t=1jαit=σkλ∫01(ln⁡1y)-(k/λ)-1[1-y]Sijy-1dy=σkλ∑m=0Sij(Sijm)(-1)Sij-m∫01(ln⁡1y)-(k/λ)-1ySij-m-1dy.
Using
(3.16)∫01(ln⁡1y)θ-1yν-1dy=1νθΓ(θ),
see (4.272.6) in Gradshteyn and Ryzhik [17]
(3.17)∴∫01(ln⁡1y)-(k/λ)-1ySij-m-1dy=1(Sij-m)-k/λΓ(-kλ);k≠0,λ,2λ,3λ,….
Substituting (3.17) in (3.15) and making some arrangements, we get (3.5).Tables1 and 2 give the values of  Ij in (3.3) and (3.5), for  j=1,2,3 and sample size n=3.Table 1
Ij(k) using (3.3) when n=3.
jIj(k)1I1(k)=kσkΓ(-k/λ)λ(∑m1=0α1(-1)m1Γ(α1+1)m1!Γ(α1-m1+1)(m1)k/λ+∑m1=0α2(-1)m1Γ(α2+1)m1!Γ(α2-m1+1)(m1)k/λ+∑m1=0α3(-1)m1Γ(α3+1)m1!Γ(α3-m1+1)(m1)k/λ)2I2(k)=kσkΓ(-k/λ)λ(∑m1=0α1∑m2=0α2(-1)m1+m2Γ(α1+1)Γ(α2+1)m1!m2!Γ(α1-m1+1)Γ(α2-m2+1)(m1+m2)k/λ+∑m1=0α1∑m2=0α3(-1)m1+m2Γ(α1+1)Γ(α3+1)m1!m2!Γ(α1-m1+1)Γ(α3-m2+1)(m1+m2)k/λ+∑m1=0α2∑m2=0α3(-1)m1+m2Γ(α2+1)Γ(α3+1)m1!m2!Γ(α2-m1+1)Γ(α3-m2+1)×(m1+m2)k/λkσkΓ(-k/λ)λ)3I3(k)=kσkΓ(-k/λ)λ(∑m1=0α1∑m2=0α2∑m3=0α3(-1)m1+m2+m3Γ(α1+1)Γ(α2+1)Γ(α3+1)m1!m2!m3!Γ(α1-m1+1)Γ(α2-m2+1)Γ(α2-m2+1)×(m1+m2+m3)k/λkσkΓλ)Table 2
Ij(k) using (3.5) when n=3.
jIj(k)1I1(k)=kσkΓ(-(k/λ))λ∑i=1n∑m=0αi(αim)(-1)αi-m(αi-m)k/λ2I2(k)=kσkΓ(-k/λ)λ(kσkΓλ∑m=0α1+α2(α1+α2m)(-1)α1+α2-m(α1+α2-m)k/λ+∑m=0α1+α3(α1+α3m)(-1)α1+α3-m(α1+α3-m)k/λ+∑m=0α2+α3(α2+α3m)(-1)α2+α3-m(α2+α3-m)k/λkσkΓλ)3kσkΓ(-k/λ)λ(kσkΓλ∑m=0α1+α2+α3(α1+α2+α3m)(-1)α1+α2+α3-m(α1+α2+α3-m)k/λ(α1+α2+α3m))Corollary 3.3.
For inid EF r.v.’s, when j  = 1, (3.3) becomes
(3.18)I1(k)=kσkΓ(-k/λ)λ∑i=1n∑r=1αi(-1)rαi(αi-1)⋯(αi-(r-1))rk/λr!.Proof.
One has(3.19)I1(k)=kσkΓ(-k/λ)λ∑i=1n(∑m1=0αiξimtk/λ)=kσkΓ(-k/λ)λ∑i=1n(∑m1=0αi(-1)m1Γ(αi+1)m1!Γ(αi-m1+1)mtk/λ)=kσkΓ(-k/λ)λ×∑i=1n(0-Γ(αi+1)Γ(αi)+Γ(αi+1)2Γ(αi-1)2k/λ-Γ(αi+1)3!Γ(αis-2)3k/λ+Γ(αi+1)4!Γ(αi-2)4k/λ+⋯+(-1)αiΓ(αi+1)αi!Γ(1)αik/λ)=kσkΓ(-k/λ)λ×∑i=1n(-αi+αi(αi-1)22k/λ-αi(αi-1)(αi-2)3!3k/λ+αi(αi-1)(αi-2)(αi-3)4!4k/λ+⋯+(-1)αiαik/λαi(αi-1)2)=kσkΓ(-k/λ)λ∑i=1n(∑r=1α(-1)r+αi(αi-1)+⋯+(αi-(r-1)rk/λr!)).
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, theIj(k)in Theorem 3.2 simply reduces to
(3.20)Ij(k)=kσkΓ(-k/λ)λ(nj)∑m1=0α⋯∑mj=0αξij(∑t=1jmt)k/λ,
where
(3.21)ξj=∏t=1j(-1)mtΓ(α+1)mt!Γ(α-mt+1),
or
(3.22)Ij(k)=kσkΓ(-k/λ)λ(nj)∑m=0Sj(Sjm)(-1)Sj-m(Sj-m)k/λ,
where
(3.23)Sj=jα.Corollary 3.5.
For iid EF r.v.’s, when j  = 1, (3.18) becomes
(3.24)I1(k)=knσkΓ(-k/λ)λ∑r=1α(-1)rα(α-1)⋅⋅⋅(α-(r-1))rk/λr!.Numerical Applications
The following examples are computed whenk=1.Example 3.6.
Letn=2 and α=2,3,4, and 5. Table 3 shows the results of the calculations.
For example, whenα=3, σ=1, λ=2,
(3.25)μ2:2=I1-I2,I1=2Γ(-1/2)2∑r=13(-1)rα(α-1)⋅⋅⋅(α-(r-1))rk/λr!=π(-α+2α(α-1)2-2α(α-1)(α-2)6)=1.73491I2=Γ(-1/2)2(22)∑m1=03∑m2=03∏t=12(-1)tΓ(α+1)mj!Γ(α-mj+1)(∑t=12mt)1/2=0.699642μ2:2=1.73491-0.699642=1.03527.Table 3
The momentsμ2:2 arising from iid Exponentiated Frechet r.v.’s. using (3.3) or (3.5).
α2345μ2:21.29151.035270.9168920.84626Example 3.7.
Settingn=2, σ=1, λ=2, and α1=1(1)5, α2=1(1)5 in Theorems 3.1 and 3.2, we get the results of the calculations in Table 4.
For example, whenα1=2,α2=3(3.26)μ2:2=I1-I2,I1=Γ(-1/2)λ∑i=12∑r=1∞(-1)rαi(αi-1)⋅⋅⋅(αi-(r-1))rk/λr!=Γ(-1/2)λ(∑r=1α1(-1)rα1(α1-1)⋯(α1-(r-1))rk/λr!+∑r=1α2(-1)rα2(α2-1)⋯(α2-(r-1))rk/λr!)=1.90574I2=Γ(-1/2)2∑m1=02∑m2=03(-1)m1+m2Γ(α1+1)Γ(α2+1)mt!m!Γ(α1-m1+1)Γ(α2-m2+1)(m1+m2)1/2=0.734577μ2:2=1.90574-0.734577=s1.17117.Table 4
The momentsμ2:2arising from inid Exponentiated Frechet r.v.’s. using (3.3) or (3.5).
α2α11234512.506631.943281.854851.822941.8073921.943281.29151.171161.12371.0992531.854851.171161.035270.9789050.94880841.822941.12370.9789050.9168920.88295351.807391.099250.9488080.8829530.84626


## 3.2. Case  2 (When the Shape Parameterα of EF Is Noninteger)

Whenα  is noninteger, the expansion of (1+z)a  is then written as(3.27)(1+z)a=∑m=0∞(am)zm,-1<z<1,see Abramowitz and Stegun [16] page 14. Using gamma notation, (3.27) can be written as(3.28)(1+z)a=∑m=0∞Γ(a+1)Γ(a-m+1)zmm!,-1<z<1.Equations (3.3) and (3.5) become(3.29)Ij(k)=kσkΓ(-k/λ)λ∑1≤i1<i2<⋯∑<ij≤n∑m1=0∞⋯∑mj=0∞ξij(∑t=1jmt)k/λ,
where
(3.30)ξij=∏t=1j(-1)mtΓ(αit+1)mt!Γ(αit-mt+1),
or(3.31)Ij(k)=kσkΓ(-k/λ)λ∑1≤i1<i2<⋯∑<ij≤n∑m=0∞(Sijm)(-1)Sij-m(Sij-m)k/λ,
where
(3.32)Sij=∑t=1jαit.Corollary 3.8.
For inid EF r.v.’s, when j  = 1, (3.29) becomes
(3.33)I1(k)=kσkΓ(-k/λ)λ∑i=1n∑r=1∞(-1)rαi(αi-1)⋅⋅⋅(αi-(r-1))rk/λr!.
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 Ij(k)in (3.29) simply reduces to
(3.34)Ij(k)=kσkΓ(-k/λ)λ(nj)∑m1=0∞⋯∑mj=0∞ξij(∑t=1jmt)k/λ,
where
(3.35)ξj=∏t=1j(-1)mtΓ(α+1)mt!Γ(α-mt+1),
or
(3.36)Ij(k)=kσkΓ(-k/λ)λ(nj)∑m=0∞(Sjm)(-1)Sj-m(Sj-m)k/λ,
where
(3.37)Sj=jα.Corollary 3.10.
For iid EF r.v.’s, when j  = 1, (3.33) becomes
(3.38)I1(k)knσkΓ(-k/λ)λ∑r=1∞(-1)rα(α-1)⋅⋅⋅(α-(r-1))rk/λr!.Example 3.11.
Letn=2 and α=1.5,2.5,3.5,4.5. Table 5 shows the results of calculations.Table 5
The momentsμ2:2arising from IID Exponentiated Frechet r.v.’s. using (3.29) or (3.31).
α1.52.53.54.5μ2:22.684690.6663081.302360.637316Remark 3.12.
Calculations in Example3.11 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.



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