Moments of Order Statistics from Nonidentically Distributed Three Parameters Beta typeI and Erlang Truncated Exponential Variables

Problem statement: Moments of order statistics of independent non-iden tically distributed (INID) random variables is not an easy subject to d eal with for continuous distributions. One is force d to use messy algebraic calculations (whether one us es permanents or not). This was the motivation behind this study. In this study the moments of ord er statistics arising from independent nonidentical ly distributed three parameters Beta type I distributi on and Erlang Truncated Exponential distribution were derived. Approach: We employed an easier technique established by Bara kat & Abdelkader will be referred to as (BAT). Results: The mean, the second moment and the variance of th e median and the smallest order statistics for the first distrib ut on were given for different values of the shape parameter and different sample sizes. Conclusion: The results can be used to make some inferences and used the BAT technique to derive moments of ord er statistics arising from independent nonidentically distributed for any other continuous di tribution with distribution function (cdf) in t he form: F(x) = 1-λ (x).


INTRODUCTION
The moments of order statistics (os) of Independent Non-Identically Distributed (INID) random variables (rvs) have been established in literature in two directions. The first direction was initiated by (Balakrishnan, 1994). It requires a basic relation between the probability density function (pdf) and the cumulative distribution function (cdf). This technique is referred to as Differential Equation Technique (DET). It enables one to compute all the singleand product moments of all order statistics in a simple recursive manner and the derivation of the moments depends mainly on integration by parts. The second technique was established by Barakat and Abdelkader(2003) will be referred to as (BAT). It is an easier manner to evaluate the moments of INID os but can be applied to distributions with cdf in the form: F( x) = 1-λ(x) or by using the survival function of the distribution under study. BAT cannot be used to evaluate the product moments. In this study it was our interest to use the BAT technique to get the moments of os arising from INID distributed three parameters Beta type I (Johnson et al., 1994) and Erlang Truncated Exponential (El-Alosey, 2007) and (Mohsin,2009) random variables .
The subject of os from INID rvs is discussed widely in the literature (David, 1981;Bapat and Beg, 1989;Davidand Nagaraja, 2003). (Barakat, 2002). found the limit behavior of bivariate order statistics from INID distributed random variables. (Güngor, Gokhan and Bulut 2005) expressed the multivariate order statistics by marginal Ordering of INID random vectors under discontinuous df`s.
Applications of the previous two methods are also found in literature for several continuous distributions. The DET was used by (Balakrishnan, 1994) to derive recurrence relations satisfied by single and product moments of os from INID rvsfor the Exponential and right truncated distributions. (Childs and Balakrishnan, 2006) applied DET to derive the moments of os from INID rvs for Logistic random variables. The first application of BAT was by (Barakat and Abdelkader, 2000) to weibull distribution and then the method was generalized by (Barakat and Abdelkader, 2003) and was applied to Erlang, Positive Exponential, Pareto and Laplace distributions. Abdelkader(2004) and (Abdelkader, 2008) used a closed expression for the survival function of Gamma and Beta distributions to compute the moments of os from INID rvs using BAT. Jamjoom(2006) applied it to Burr XII rvs. Let X 1 , X 2 ,…,X n be independent random variables having cdfs F 1 (x), F 2 (x),…, F n (x) and pdfs,ƒ 1 (x), ƒ 2 (x),…, ƒ n (x) respectively. Let X 1:n ≤, X 2:n ≤…≤X n:n denote the os obtained by arranging the n X i 's in increasing order of magnitude. Then the pdf of the rth os.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, 1989) put the previous pdf of the rth os X r:n in the form of permanent as: To derive the moments of os from INID rvs arising from this distribution we need the following theorem which is established by (Barakat and Abdelkader, 2003).

Proof:
The proof of this theorem can be found in Barakat and Abdelkader (2003).

MATERIALS AND METHODS
We consider the three parameters Bet type I distribution (Johnson et al., 1994) with cdf: and Erlang truncated exponential (El-Alosey, 2007) and (Mohsin, 2009) with cdf: Now, we consider the case when the variables X i , s be INID rvs, then the above cdfs can be written as: Applications of theorem 1 for the above distributions will be stated as theorem 2 nd 3.

Moments of os from INID erlang truncated exponential rvs:
Theorem 3: For 1≤r≤nand k = 1,2, …: Proof: On applying theorem 1 and using (8), we get: On integrating term by term using the integral: and the proof is completed.

RESULTS
Result 1: Substituting (9) in (3) the kth moments of the rth os from INID three parameters beta type I can be finally written as: Remark 1: The kth moment of the largest os X n:n from INID three Beta type I rvs can be written as: where, I j (k) is defined in (9).

Remark 2:
The kth moment of the first os X 1:n from INID three parameters Beta type I rvs can be written as: Remark 3: IID case The Independent Identically Distributed (IID) case can be deduced from theorem (2). Ij(k) can be written as: where, βz (a, b) is the incomplete beta function defined before.
Remark 1: The kth moment of the largest osX n:n from INID Erlang truncated exponential rvs can be written as: where, Ij (k) is defined in (10) Remark 2: The k th moment of the first os X 1:n from INID Erlang truncated exponential rvs. can be written as: Remark 3: The Independent Identically Distributed (IID) case can be deduced from theorem (2) . I j (k) is written as: Numerical calculations for three parameter beta type I: Simple programs written by Mathematica 7 were used to calculate the moments given in tables 1, 2, 3, 4, 5.

DISCUSSION
Using the BAT Technique beautiful and elegant recurrence rlations were obtained for the moments of rder statistics from nonidentically but independently distributed variables for both distributions under study which can be considered an addition to INID. os.

CONCLUSION
BAT technique is strongly recommended for any other continuous distribution with cdf in the form: F( x ) = 1-λ (x). Comparison between the DET and BAT technique is also recommended for future study.