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

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 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.

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 as𝐹(𝑥)=1−1−𝑒^{−𝜎𝑥𝜆}^𝛼,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 calculated expressions for the 𝑛th 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 the 𝑟th o.s. arising from inid EF distribution are given in Section 2.

Computation of moments of the 𝑟th 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 form 𝐹(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.

Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be independent random variables having cumulative distribution functions 𝐹₁(𝑥),𝐹₂(𝑥),…,𝐹_𝑛(𝑥) and probability density functions 𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑛(𝑥), respectively. Let 𝑋_1∶𝑛≤𝑋_2∶𝑛≤⋯≤𝑋_𝑛∶𝑛 denote the order statistics obtained by arranging the 𝑛𝑋^,_𝑖𝑠 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 as𝑓_𝑟∶𝑛(𝑥)=1(𝑟−1)!(𝑛−𝑟)!_𝑝^𝑟−1_𝑎=1𝐹_𝑖𝑎(𝑥)𝑓_𝑖𝑟(𝑥)^𝑛_𝑐_=𝑟+11−𝐹_𝑖𝑐(𝑥),(2.1) where ∑_𝑝  denotes the summation over all 𝑛! permutations (𝑖₁,𝑖₂,…,𝑖_𝑛) of (1,2,…𝑛). Bapat and Beg [6] put it in the form of the permanent as𝑓_𝑟∶𝑛(𝑥)=1(𝑟−1)!(𝑛−𝑟)!per⎡⎢⎢⎣𝐹(𝑥)_{𝑟−1}𝑓(𝑥)_1{1−𝐹(𝑥)}_{𝑛−𝑟}⎤⎥⎥⎦,𝐹_(𝑟)(𝑥)=^𝑛_𝑗=𝑟_𝑝𝑗^𝑗_𝑎=1𝐹_𝑖𝑎(𝑥)^𝑛_𝑎_=𝑗+11−𝐹_𝑖𝑎(𝑥).(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.

Figure 1 

Graph of p.d.f. of (inid) o.s.  from Ef distribution for selected values of 𝛼1=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 of 𝛼1=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,…,𝑖_𝑛) for (1,…,𝑛)which satisfy 𝑖₁<𝑖₂<⋯<𝑖_𝑗  and 𝑖_𝑗+1<𝑖_𝑗+2<⋯<𝑖_𝑛. And using the permanent, we have𝐹_(𝑟)(𝑥)=^𝑛_𝑖=𝑟1𝑖!(𝑛−𝑖)!per⎡⎢⎢⎢⎢⎢⎣𝐹₁(𝑥)1−𝐹₁(𝑥)⋮𝐹_𝑛(𝑥)_{𝑖}1−𝐹_𝑛(𝑥)_{𝑛−𝑖}⎤⎥⎥⎥⎥⎥⎦,−∞<𝑥<∞.(2.3) For EF distribution, we have𝐹_𝑖(𝑥)=1−1−𝑒^{−(𝜎/𝑥)𝜆}^𝛼𝑖,𝑥>0,𝜎>0,𝜆>0,𝛼_𝑖>0,𝑓_𝑖(𝑥)=𝛼_𝑖𝜆𝜎^𝜆+1𝑥^−(𝜆+1)1−𝑒^{−(𝜎/𝑥)𝜆}^𝛼𝑖−1𝑒^{−(𝜎/𝑥)𝜆},𝑥>0,𝜎>0,𝜆>0,𝛼_𝑖>0.(2.4)

3. The Moments of the 𝑟th 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. Let 𝑋₁,𝑋₂,…,𝑋_𝑛be 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 𝜇^(𝑘)_𝑟∶𝑛=^𝑛_{𝑗=𝑛−𝑟+1}(−1)^{𝑗−(𝑛−𝑟+1)}⎛⎜⎜⎝𝑗−1𝑛−𝑟⎞⎟⎟⎠𝐼_𝑗(𝑘),(3.1) where 𝐼_𝑗(𝑘)=_{1≤𝑖1<𝑖2<⋯<𝑖𝑗≤𝑛}…𝑘^∞₀𝑥^𝑘−1𝑗_𝑡=1𝐺_𝑖𝑡(𝑥)𝑑𝑥,𝑗=1,2,…,𝑛,(3.2)𝐺_𝑖𝑡(𝑥)=1−𝐹_𝑖𝑡(𝑥), with (𝑖₁,𝑖₂,…,𝑖_𝑛) are permutations of (1,2,…,𝑛) for which 𝑖₁≤𝑖₂<⋯<𝑖_𝑛.
The following theorem gives an explicit expression for 𝐼_𝑗(𝑘)when 𝑋₁,𝑋₂,…,𝑋_𝑛are inid EF r.v.’s. Two cases were considered.

3.1. Case  1 (When The Shape Parameter 𝛼 of EF Distribution Is an Integer)

Theorem 3.2. For 1≤𝑟≤𝑛,𝑘=1,2,…, 𝐼_𝑗(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}^𝛼𝑖1_𝑚1=0⋯^𝛼𝑖𝑗_𝑚𝑗=0𝜉_𝑖𝑗⎛⎜⎝^𝑗_𝑡=1𝑚_𝑡⎞⎟⎠^𝑘/𝜆,(3.3) where 𝜉_𝑖𝑗=^𝑗_𝑡₌₁(−1)^𝑚𝑡Γ𝛼_𝑖𝑡+1𝑚_𝑡!Γ𝛼_𝑖𝑡−𝑚_𝑡+1,(3.4) or 𝐼_𝑗(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}^𝑆𝑖𝑗_𝑚=0⎛⎜⎜⎝𝑆_𝑖𝑗𝑚⎞⎟⎟⎠(−1)^{𝑆𝑖𝑗−𝑚}𝑆_𝑖𝑗−𝑚^𝑘/𝜆,(3.5) where 𝑆_𝑖𝑗=^𝑗_𝑡=1𝛼_𝑖𝑡.(3.6)

Proof. On applying Theorem 3.1 and using (2.4), we get 𝐼_𝑗(𝑘)=_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}𝑘^∞₀𝑥^𝑘−1𝑗_𝑡=11−𝑒^{−(𝜎/𝑥)𝜆}^𝛼𝑖𝑡𝑑𝑥.(3.7) This integral converges if ∑^𝑗_𝑡=1𝛼_𝑖𝑡>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 (𝑏+𝑧)^𝑎=^𝑎_𝑚₌₀⎛⎜⎜⎝𝑎𝑚⎞⎟⎟⎠𝑏^𝑎−𝑚𝑧^𝑚,(3.8) see Abramowitz and Stegun [16] page 10.
Using the Gamma notation and considering 𝑏=1, this series can be written as (1+𝑧)^𝑎=^𝑎_𝑚₌₀Γ(𝑎+1)Γ(𝑎−𝑚+1)𝑧^𝑚𝑚!.(3.9)
Then the bracket in (3.7) can be written as 1−𝑒^{−(𝜎/𝑥)𝜆}^𝛼𝑖𝑡=^𝛼𝑖𝑡_𝑚=0Γ𝛼_𝑖𝑡+1Γ𝛼_𝑖𝑡−𝑚+1(−1)^𝑚𝑒^{−𝑚(𝜎/𝑥)𝜆}𝑚!∴𝐼_𝑗(𝑘)=_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}𝑘^∞₀𝑥^𝑘−1𝑗_𝑡=1^𝛼𝑖𝑡_𝑚=0Γ𝛼_𝑖𝑡+1Γ𝛼_𝑖𝑡−𝑚+1(−1)^𝑚𝑒^{−𝑚(𝜎/𝑥)𝜆}𝑚!𝑑𝑥=_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}𝑘^∞₀𝑥^𝑘−1⎛⎜⎜⎜⎝^𝛼𝑖1_𝑚₁₌₀Γ𝛼_𝑖1+1Γ𝛼_𝑖1−𝑚₁+1(−1)^𝑚1𝑒^{−𝑚1(𝜎/𝑥)𝜆}𝑚₁!⋯^𝛼𝑖𝑗_𝑚𝑗=0Γ𝛼_𝑖𝑡+1Γ𝛼_𝑖𝑗−𝑚_𝑗+1×(−1)^𝑚𝑗𝑒^{−𝑚𝑗(𝜎/𝑥)𝜆}𝑚_𝑗!⎞⎟⎟⎟⎠𝑑𝑥∴𝐼_𝑗(𝑘)=_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}𝑘^∞₀𝑥^𝑘−1⎛⎜⎜⎜⎝^𝛼𝑖1_𝑚₁₌₀⋯^𝛼𝑖𝑗_𝑚𝑗=0Γ𝛼_𝑖1+1Γ𝛼_𝑖1−𝑚₁+1⋯Γ𝛼_𝑖𝑗+1Γ𝛼_𝑖𝑗−𝑚_𝑗+1×(−1)^{∑𝑗𝑡=1𝑚𝑡}𝑒^{−∑𝑗𝑡=1𝑚𝑡(𝜎/𝑥)𝜆}_𝑗∏_𝑡=1𝑚_𝑡!⎞⎟⎟⎟⎠𝑑𝑥=_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}𝑘^𝛼𝑖1_𝑚1=0⋯^𝛼𝑖𝑗_𝑚𝑗=0⎛⎜⎝^𝑗_𝑡=1Γ𝛼_𝑖𝑡+1Γ𝛼_𝑖𝑡−𝑚_𝑡+1(−1)^𝑚𝑡𝑚_𝑡!×^∞₀𝑥^𝑘−1𝑒^{−∑𝑗𝑡=1𝑚𝑡(𝜎/𝑥)𝜆}𝑑𝑥=_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}𝑘^𝛼𝑖1_𝑚1=0⋯^𝛼𝑖𝑗_𝑚𝑗=0⎛⎜⎝^𝑗_𝑡=1(−1)^𝑚𝑡𝑚_𝑡!Γ𝛼_𝑖𝑡+1Γ𝛼_𝑖𝑡−𝑚_𝑡+1×^∞₀𝑥^𝑘−1𝑒^{−∑𝑗𝑡=1𝑚𝑡(𝜎/𝑥)𝜆}𝑑𝑥.(3.10) Substituting 𝑦=^𝑗_𝑡=1𝑚_𝑡𝜎𝑥^𝜆∴𝑥=𝜎⎛⎜⎝^𝑗_𝑡=1𝑚_𝑡⎞⎟⎠^1/𝜆𝑦^−1/𝜆𝑑𝑥=𝜎𝜆⎛⎜⎝^𝑗_𝑡=1𝑚_𝑡⎞⎟⎠^1/𝜆𝑦^{−(1/𝜆)−1}𝑑𝑦∴^∞₀𝑥^𝑘−1𝑒^{−∑𝑗𝑡=1𝑚𝑡(𝜎/𝑥)𝜆}𝑑𝑥=𝜎^𝑘⎛⎜⎝^𝑗_𝑡=1𝑚_𝑡⎞⎟⎠^𝑘/𝜆^∞₀𝑦^{−(𝑘/𝜆)−1}𝑒^−𝑦𝑑𝑦=𝜎^𝑘𝜆⎛⎜⎝^𝑗_𝑡=1𝑚_𝑡⎞⎟⎠^𝑘/𝜆Γ−𝑘𝜆;𝜆>0,𝑘=1,2,…,𝑘𝜆≠0,1,2,….(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) ∴𝑥=𝜎ln1𝑦^−1/𝜆,(3.13) Then 𝑑𝑥=−𝜎𝜆ln1𝑦^{−(1/𝜆)−1}𝑦⁻¹𝑑𝑦∴^∞₀𝑥^𝑘−1𝑗_𝑡=11−𝑒^{−(𝜎/𝑥)𝜆}^𝛼𝑖𝑡𝑑𝑥=^∞₀𝑥^𝑘−11−𝑒^{−(𝜎/𝑥)𝜆}^𝑆𝑖𝑗𝑑𝑥,(3.14) where 𝑆_𝑖𝑗=^𝑗_𝑡=1𝛼_𝑖𝑡=𝜎^𝑘𝜆¹₀ln1𝑦^{−(𝑘/𝜆)−1}[1−𝑦]^𝑆𝑖𝑗𝑦⁻¹𝑑𝑦=𝜎^𝑘𝜆^𝑆𝑖𝑗_𝑚=0⎛⎜⎜⎝𝑆_𝑖𝑗𝑚⎞⎟⎟⎠(−1)^{𝑆𝑖𝑗−𝑚}¹₀ln1𝑦^{−(𝑘/𝜆)−1}𝑦^{𝑆𝑖𝑗−𝑚−1}𝑑𝑦.(3.15) Using ¹₀ln1𝑦^𝜃−1𝑦^𝜈−1𝑑𝑦=1𝜈^𝜃Γ(𝜃),(3.16) see (4.272.6) in Gradshteyn and Ryzhik [17] ∴¹₀ln1𝑦^{−(𝑘/𝜆)−1}𝑦^{𝑆𝑖𝑗−𝑚−1}𝑑𝑦=1𝑆_𝑖𝑗−𝑚^−𝑘/𝜆Γ−𝑘𝜆;𝑘≠0,𝜆,2𝜆,3𝜆,….(3.17) Substituting (3.17) in (3.15) and making some arrangements, we get (3.5).

Tables 1 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 

𝐼_𝑗(𝑘) using (3.3) when 𝑛=3.

Table 2 

𝐼_𝑗(𝑘) using (3.5) when 𝑛=3.

Corollary 3.3. For inid EF r.v.’s, when j = 1, (3.3) becomes 𝐼₁(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆^𝑛_𝑖=1^𝛼𝑖_𝑟=1(−1)^𝑟𝛼_𝑖𝛼_𝑖−1⋯𝛼_𝑖−(𝑟−1)𝑟^𝑘/𝜆𝑟!.(3.18)

Proof. One has 𝐼₁(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆^𝑛_𝑖=1⎛⎜⎝^𝛼𝑖_𝑚₁₌₀𝜉_𝑖𝑚_𝑡^𝑘/𝜆⎞⎟⎠=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆^𝑛_𝑖=1⎛⎜⎝^𝛼𝑖_𝑚₁₌₀(−1)^𝑚1Γ𝛼_𝑖+1𝑚₁!Γ𝛼_𝑖−𝑚₁+1𝑚_𝑡^𝑘/𝜆⎞⎟⎠=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆×^𝑛_𝑖=1⎛⎜⎝0−Γ𝛼_𝑖+1Γ𝛼_𝑖+Γ𝛼_𝑖+12Γ𝛼_𝑖−12^𝑘/𝜆−Γ𝛼_𝑖+13!Γ𝛼_𝑖𝑠−23^𝑘/𝜆+Γ𝛼_𝑖+14!Γ𝛼_𝑖−24^𝑘/𝜆+⋯+(−1)^𝛼𝑖Γ𝛼_𝑖+1𝛼_𝑖!Γ(1)𝛼_𝑖^𝑘/𝜆⎞⎟⎠=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆×^𝑛_𝑖=1⎛⎜⎝−𝛼_𝑖+𝛼_𝑖𝛼_𝑖−122^𝑘/𝜆−𝛼_𝑖𝛼_𝑖−1𝛼_𝑖−23!3^𝑘/𝜆+𝛼_𝑖𝛼_𝑖−1𝛼_𝑖−2𝛼_𝑖−34!4^𝑘/𝜆+⋯+(−1)^𝛼𝑖𝛼_𝑖^𝑘/𝜆⎞⎟⎠=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆^𝑛_𝑖=1⎛⎜⎝^𝛼_𝑟=1(−1)^𝑟+𝛼_𝑖𝛼_𝑖−1+⋯+𝛼_𝑖−(𝑟−1)𝑟^𝑘/𝜆𝑟!⎞⎟⎠.(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 𝐼_𝑗(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠^𝛼_𝑚₁₌₀⋯^𝛼_𝑚𝑗=0𝜉_𝑖𝑗⎛⎜⎝^𝑗_𝑡=1𝑚_𝑡⎞⎟⎠^𝑘/𝜆,(3.20) where 𝜉_𝑗=^𝑗_𝑡=1(−1)^𝑚𝑡Γ(𝛼+1)𝑚_𝑡!Γ𝛼−𝑚_𝑡+1,(3.21) or 𝐼_𝑗(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠^𝑆𝑗_𝑚₌₀⎛⎜⎜⎝𝑆_𝑗𝑚⎞⎟⎟⎠(−1)^{𝑆𝑗−𝑚}𝑆_𝑗−𝑚^𝑘/𝜆,(3.22) where 𝑆_𝑗=𝑗𝛼.(3.23)

Corollary 3.5. For iid EF r.v.’s, when j = 1, (3.18) becomes 𝐼₁(𝑘)=𝑘𝑛𝜎^𝑘Γ(−𝑘/𝜆)𝜆^𝛼_𝑟=1(−1)^𝑟𝛼(𝛼−1)⋅⋅⋅(𝛼−(𝑟−1))𝑟^𝑘/𝜆𝑟!.(3.24)

Numerical Applications
The following examples are computed when 𝑘=1.

Example 3.6. Let 𝑛=2 and 𝛼=2,3,4, and 5. Table 3 shows the results of the calculations.
For example, when 𝛼=3, 𝜎=1, 𝜆=2, 𝜇_2∶2=𝐼₁−𝐼₂,𝐼₁=2Γ(−1/2)2³_𝑟=1(−1)^𝑟𝛼(𝛼−1)⋅⋅⋅(𝛼−(𝑟−1))𝑟^𝑘/𝜆𝑟!=√𝜋⎛⎜⎝−𝛼+√2𝛼(𝛼−1)2−√2𝛼(𝛼−1)(𝛼−2)6⎞⎟⎠=1.73491𝐼₂=Γ(−1/2)2⎛⎜⎜⎝22⎞⎟⎟⎠³_𝑚1=0³_𝑚2=0²_𝑡=1(−1)^𝑡Γ(𝛼+1)𝑚_𝑗!Γ𝛼−𝑚_𝑗+1⎛⎜⎝²_𝑡=1𝑚_𝑡⎞⎟⎠^1/2=0.699642𝜇_2∶2=1.73491−0.699642=1.03527.(3.25)

Table 3 

The moments 𝜇_2∶2 arising from iid Exponentiated Frechet r.v.’s. using (3.3) or (3.5).

Example 3.7. Setting 𝑛=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, when 𝛼₁=2,𝛼₂=3𝜇_2∶2=𝐼₁−𝐼₂,𝐼₁=Γ(−1/2)𝜆²_𝑖=1^∞_𝑟=1(−1)^𝑟𝛼_𝑖𝛼_𝑖−1⋅⋅⋅𝛼_𝑖−(𝑟−1)𝑟^𝑘/𝜆𝑟!=Γ(−1/2)𝜆⎛⎜⎝^𝛼1_𝑟=1(−1)^𝑟𝛼₁𝛼₁−1⋯𝛼₁−(𝑟−1)𝑟^𝑘/𝜆𝑟!+^𝛼2_𝑟=1(−1)^𝑟𝛼₂𝛼₂−1⋯𝛼₂−(𝑟−1)𝑟^𝑘/𝜆𝑟!=1.90574𝐼₂=Γ(−1/2)2²_𝑚1=0³_𝑚2=0(−1)^𝑚1+𝑚2Γ𝛼₁+1Γ𝛼₂+1𝑚_𝑡!𝑚!Γ𝛼₁−𝑚₁+1Γ𝛼₂−𝑚₂+1𝑚₁+𝑚₂^1/2=0.734577𝜇_2∶2=1.90574−0.734577=𝑠1.17117.(3.26)

Table 4 

The moments 𝜇_2∶2arising from inid Exponentiated Frechet r.v.’s. using (3.3) or (3.5).

3.2. 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(1+𝑧)^𝑎=^∞_𝑚₌₀⎛⎜⎜⎝𝑎𝑚⎞⎟⎟⎠𝑧^𝑚,−1<𝑧<1,(3.27)

see Abramowitz and Stegun [16] page 14. Using gamma notation, (3.27) can be written as(1+𝑧)^𝑎=^∞_𝑚₌₀Γ(𝑎+1)Γ(𝑎−𝑚+1)𝑧^𝑚𝑚!,−1<𝑧<1.(3.28)

Equations (3.3) and (3.5) become𝐼_𝑗(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}^∞_𝑚1=0⋯^∞_𝑚𝑗=0𝜉_𝑖𝑗⎛⎜⎝^𝑗_𝑡=1𝑚_𝑡⎞⎟⎠^𝑘/𝜆,(3.29) where 𝜉_𝑖𝑗=^𝑗_𝑡=1(−1)^𝑚𝑡Γ𝛼_𝑖𝑡+1𝑚_𝑡!Γ𝛼_𝑖𝑡−𝑚_𝑡+1,(3.30) or𝐼_𝑗(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆_{1≤𝑖1<𝑖2<⋯}_{<𝑖𝑗≤𝑛}^∞_𝑚=0⎛⎜⎜⎝𝑆_𝑖𝑗𝑚⎞⎟⎟⎠(−1)^{𝑆𝑖𝑗−𝑚}𝑆_𝑖𝑗−𝑚^𝑘/𝜆,(3.31) where 𝑆_𝑖𝑗=^𝑗_𝑡=1𝛼_𝑖𝑡.(3.32)

Corollary 3.8. For inid EF r.v.’s, when j = 1, (3.29) becomes 𝐼₁(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆^𝑛_𝑖=1^∞_𝑟=1(−1)^𝑟𝛼_𝑖𝛼_𝑖−1⋅⋅⋅𝛼_𝑖−(𝑟−1)𝑟^𝑘/𝜆𝑟!.(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 𝐼_𝑗(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠^∞_𝑚₁₌₀⋯^∞_𝑚𝑗=0𝜉_𝑖𝑗⎛⎜⎝^𝑗_𝑡=1𝑚_𝑡⎞⎟⎠^𝑘/𝜆,(3.34) where 𝜉_𝑗=^𝑗_𝑡=1(−1)^𝑚𝑡Γ(𝛼+1)𝑚_𝑡!Γ𝛼−𝑚_𝑡+1,(3.35) or 𝐼_𝑗(𝑘)=𝑘𝜎^𝑘Γ(−𝑘/𝜆)𝜆⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠^∞_𝑚₌₀⎛⎜⎜⎝𝑆_𝑗𝑚⎞⎟⎟⎠(−1)^{𝑆𝑗−𝑚}𝑆_𝑗−𝑚^𝑘/𝜆,(3.36) where 𝑆_𝑗=𝑗𝛼.(3.37)

Corollary 3.10. For iid EF r.v.’s, when j = 1, (3.33) becomes 𝐼₁(𝑘)𝑘𝑛𝜎^𝑘Γ(−𝑘/𝜆)𝜆^∞_𝑟=1(−1)^𝑟𝛼(𝛼−1)⋅⋅⋅(𝛼−(𝑟−1))𝑟^𝑘/𝜆𝑟!.(3.38)

Example 3.11. Let 𝑛=2 and 𝛼=1.5,2.5,3.5,4.5. Table 5 shows the results of calculations.

Table 5 

The moments 𝜇_2∶2arising from IID Exponentiated Frechet r.v.’s. using (3.29) or (3.31).

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.