Computing the Moments of Order Statistics from Independent Non – Identically Distributed Burr Type XII Random Variables

In this paper, we derive a recurrence relation for computing all single moments of all order statistics arising from independent but not identically distributed Burr type XII random variables.

In the last few years much attention had been paid to order statistics from independent nonidentically distributed variables i.ni.d [5,6,7,8] .
Derivation of recurrence relations for single moments of order statistics from i.ni.d available samples found in the literature have taken two directions, the work initiated by Balakrishnan [5,6] and that of Barakat and Abdelkader [9] In Balakrishnan 's work [5,6] , a linear relation between the PDF and CDF of the distribution , if exists, is exploited and then one has to go through messy calculations using integration by parts to get the result. Application of this method was done on many distributions such as: exponential [5] , right-truncated exponential [6] doublytruncated exponential and logistic distribution [10,11] , power function distribution [7] , Pareto and doublytruncated Pareto distributions [12] . All of these results were obtained by exploiting a basic differential equation satisfied by the distribution under consideration. For example: the differential equation satisfied by the PDF and CDF of exponential distribution is and for power function distributions it is However, most of these recurrence relations show that it is enough to evaluate the kth moment of a single order statistics in a sample of size n, if these moments in samples of size less than n are already available. The kth moments of the remaining n -1 order statistics can then be determined by repeated use of these recurrence relations.
Barakat and Abdelkader [9] generalized their procedure initiated in (2000) [13] to any d.f. and expressed the kth moment of the rth order statistics of a sample of size n purely in terms of the kth moments of the maximum order statistics or of the minimum order statistics from samples of size up to n of all possible subsamples of the given samples. This in fact simplifies the recursive computation of the single moments of (i.nid) order statistics.
Application of Barakat and Abdelkader 's method [9] started in fact in (2000) [13] when they first applied it to calculate single moments of non-identically distributed Weibull random and in the year (2004) to Erlang distribution by [14] . The advantages of their procedure can be simply described as follows: first there is no conditions imposed on the CDF. and PDF. of the underlying distribution, i.e. whether they are related on not ; obtained by their method is purely expressed in terms of the kth moments of maximums and the minimums of all possible subsamples of the given sample.
In this paper we consider the case where the r.v.s n i i X , ... , 2 , 1 , = are independent and non identical having Burr type XII distribution with CDF.
In the next section we derive the kth moment of the largest order statistics

Proof
Since , then by definition of moments we have: This equation was obtained by Galambos [36] . Then the kth moment of the smallest is ( ) x n F is the CDF of the smallest order statistics from independent not identically distributed random variables defined by ( ) Now substituting (2) in (11) expanding the summation on the first term, then where the summation P extends over all permutation ( )      Generalization of this table is mentioned in [13] .