Moments of order statistics from nonidentically Distributed Lomax, exponential Lomax and exponential Pareto Variables

In this paper, the probability density function and the cumulative distribution function of the r th order statistic arising from independent nonidentically distributed (INID) Lomax, exponential Lomax and exponential Pareto variables are presented. The moments of order statistics from INID Lomax, exponential lomax and exponential Pareto were derived using the technique established by Barakat and Abdelkader. Also, numerical examples are given.


Introduction
Three techniques have been established in literature to compute moments of order statistics of independent nonidentically distributed random variables. The first technique is created by Balakrishnan (1994). This technique requires a basic relation between to probability density function (pdf) and the cumulative distribution function (cdf) and referred to as differential equation technique (DET). Many authors used to this technique to derive the moments of INID order statistics for continuous distributions (See, Childs and Balakrishnan (2006) and Mohie Elidin et al (2007)). Barakat and Abdelkader (2000) established the second technique to compute the moments of order statistics from nonidentically distributed weibull variables and referred to as (BAT). This technique that the cdf of the distribution can be written in the form F(x) = 1 - (x), which is satisfied in this distribution. Many authors used to BAT technique to compute the moments of INID order statistics for several continuous distributions (more details, see Abdelkader (2004a) Abdelkader (2004b), Jamjoom (2006), Abdelkader (2010), Jamjoom and Al-Saiary (2010) and Jamjoom and Al-Saiary (2013)). The third technique was developed by Jamjoom and Al-Saiary (2011) which is the moment generating function technique it depends on (BAT) and referred to as (M.G.F. BAT). The third technique was used by Al-Saiary (2015) to compute the moments of INID order statistics for standard type II generalized logistic variables. In this paper the pdf and the cdf of the r th order statistic arising from INID Lomax, exponential Lomax and exponential Pareto distributions are given in section 2. In section 3, we drive the moments of the r th order statistics of INID random variables arising from Lomax, exponential Lomax and exponential Pareto using (BAT). Finally, some conclusions are addressed in section 4.

Pareto distributions
Let X 1 , X 2 , …, X n be independent random variables having cumulative distribution functions (cdfs) F 1 (x), …, F n (x) and probability density functions (pdfs) f 1 (x), f 2 (x), …, f n (x), respectively. Let x 1:n  …  x n:n denote the order statistics obtained by arranging the n in increasing order of magnitude. Then the pdf of the r th order statistic X r:n , (1  r  n), can be written as: (1) Where denotes the summation overall n! permutations (i 1 , i 2 , …, i n ) of (1,2,…,n) (See Bapat and Beg (1989)). Put the previous pdf of the r th order statistic X r:n in the form of permanent as: 3 4 7 -1921 V o l u m e 1 4 N u m b e r 1 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 7432 | P a g e J a n u a r y 2018 w w w . c i r w o r l d . c o m (2) and the cdf of the r th order statistic X r:n can be written as: Where P j is all permutations of i 1 , i 2 , …, i n for 1, 2, …, n which satisfy i 1 < i 2 < … < i j and i j+1 < i j+2 < … < i n and it is conveniently expressed in terms of permanent as (4) Where F(x) and [1 -  respectively. So the F r:n (x) can be rewritten as (5) The cdf of the smallest order statistic (r = 1) X 1:n is given by: and the cdf of the largest order statistic (r = n) X n:n (x) is given by: Note that permanent is a square matrix, which is defined similar to the determinants except that all elements in the expansion have a positive sign (Minc (1987) and Balakrishnan (1994)).
In this paper, we consider the case where the random variables X i , i = 1, 2, …, n are independent and nonidentical having Lomax (L), exponential Lomax (EL) and exponential Pareto (EP) distributions with cdfs, F L (x), F EL (x) and F EP (x) and pdfs f L (x), f EL (x) and f EP (x) respectively (more details see, Lemonte and Corderio (

The moment of the r th order statistics arising from INID Lomax, exponential lomax and exponential Pareto random variables
In this section, we drive the moments of order statistics from INID random variables arising from L, EL and EP. We need to following theorem which is established by Barakat and Abdelkader (2003).
Theorem 1: Let X 1 , X 2 , …, X n be independent nonidentically distribution random variables, The k th moment of order statistics for 1  r  n and k = 1, 2, … is given by: Where (10) with (i 1 , i 2 , …, i n ) are permutations of (1, 2, …, n) for which i 1 < i 2 < … < i n . The proof of this Theorem 1 see to Barakat and Abdelkader (2003).

Moments of order statistics from INID Lomax random variables
The following Theorem gives an explicit expression for I j (k) when X 1 , X 2 , …, X n are INID Lomax random variables.

Moments of order statistic from INID exponential Lomax random variables
Proof: By using Theorem 1 and eq. (10), then  (9), the k th moments of the r th order statistics from INID exponential Lomax can be written as: (18) Also, the k th moment of the smallest order statistic x 1:n from INID exponential Lomax random variables can be written as: (19) and the k th moment of the largest order statistic x n:n from INID exponential Lomax random variables can be written as: (20)

Conclusion
In this paper, exact moments of the r th order statistic form independent and nonidentically distributed random variables for the Lomax, exponential Lomax and exponential Pareto distribution are derived using the BAT technique. Some numerical examples are presented to illustrate the theorems and results of this study.