Exact Moments of order Statistics from The exponentiated Lomax distribution

In this paper, order statistics from the exponentiated Lomax distribution (ELD) are obtained. Exact form for the single, product and Triple moment of order statistics from ELD are derived. Measures of skewness and kurtosis of the probability density function of the r th order statistic are presented. Some recurrence relations for the single and product moments of order statistics from ELD are established. Also, the percentage points of single order statistics from ELD are computed.


Introduction
Order statistics have been used in many applications, including estimation and detection of extreme values, quality control, goodness of fit, robust statistical, analysis of censored sample, etc (more details see Balakirshnan and Choen 1991, Balakirshnan and Chan 1998, David andNagaraja 2003 andTaher et al 2015).
The moments of order statistics have some important applications in inferential methods. Several authors have studied the probability density function and the moments of order statistics, in addition to the derivation of some recurrence relations of these moments arising from many specific continuous distributions such as pareto, exponential, gamma, logistic, half logistic, Burr Type X, exponentiated Log-logestic and Poisson-Lomax. (more details, see Malik 1966 , x > 0, ,  and  > 0 (1) and the cumulative distribution function (cdf) of X is given by F (x; , , ) = [1 -(1 + x) - ]  , x > 0, ,  and  > 0 (2) Where  and  are the shape parameters and  is the scale parameter. The survival function S(x), hazard rate function h(x), reversed hazard rate function r(x) and the cumulative hazard rate function H (x) of ELD are given by S (x; , , ) = 1 -F (x; , , ) = 1 -  (6) Note that, when =1, the pdf of the ELD reduces to Lomax distribution, =1, the pdf of the ELD reduces, to exponentiated pareto distribution and ==1 reduce to standard pareto distribution.
In this paper, we obtain exact form expressions for the pdf of order statistics for ELD in section 2. In section 3, we derive exact expressions for the single, product and triple moments for order statistics from ELD and compute the measures of skewness and kurtosis of the pdf of the r th order statistics. We establish some recurrence relations for the single and Where with B (a,b) being the complete beta function and f (x) = f (x; , , ) and F (x) = F (x; , , ) are pdf and cdf given in (1) and (2).
Theorem 2.1. Let f (x) and F (x) be the pdf and cdf of ELD for a random variable X. Then the pdf of the r th order statistic say f r:n (x) is given by (8) Proof: by using binomial expansion, the pdf in (7) can be written as Let Then Note that, d i (n,r), (i = 0, 1, 2,…,n-r), are coefficients not dependent on ,  and . This observation means that f r:n (x) is a weighted average of exponentiated lomax densities. As special cases of (8), the pdf of the smallest (r=1) and the largest (r=n) order statistics can be easily obtained as and f n:n (x) = f (x; n, , ) x > 0 The joint pdf of any two order statistics x = x r:n and y = x s:n for 1  r < s  n is given by f r,s:n (x,y) = C r,s: n-s f(x) f(y), 0 < x < y <  (9) , 2. Let X r:n and X s:n for 1  r < s  n be the r th and s th order statistics for the ELD. Then the joint pdf of X r:n and X s:n is given by proof: By using binomial expansion, the pdf in (9) can be written as By substituting (1) and (2) in the previous equation, then , 0 < x < y <  × f (y;  (s-r-i+j), , ) , 0 < x < y <  Using the same method that used to prove theories (2.1) and (2.2), we can get the joint pdf of three order statistics as follows: Theorem 2.3. Let X r:n , X s:n and X t:n for 1  r < s < t  n be the r th , s th and t th order statistics from the ELD. Then the joint pdf of X r:n , X s:n and X t:n is given by f r,s,t:n (x,y,z) = C r,s,t:n [F(x)] Where By substituting (1) and (2)

Moments of order statistics
In this section, we derive a closed form expressions for the single, product and triple moments for order statistics from ELD.

Single moments
Theorem 3.1. Let x 1 , x 2 ,…, x n be a random sample of size n from the ELD and let x 1:n , x 2:n ,…, x n:n denote the corresponding order statistics. Then the k th moment of the r th order statistic for k = 1, 2,…, denoted by is given by (13) Proof: we know that By using binomial expansion, then Also, we can prove that for n = r, formula in (13) reduces to and r = n, formula in (13) reduces to Moreover, the measures of skewness (sk) and kurtosis (ku) of the distribution of the r th order statistic can be computed from the following equations (15) and (16) Note that the variance of X r:n = The following table (1) shows the values of sk and ku of X r:n for different values of  = 5, 7,  = 2, 4 and  = 2, 3 when n = 2, 3, 5 and 1  r  n.    (1), for these selected values, we notice that the distribution of the r th order statistic is positively skewed because all values of sk more than zero. Also, the distribution of the r th order statistic is leptokurtic (higher and sharper peaked than the normal distribution) because all values of ku more than 3.

Product moments
Theorem 3.2. for the exponentiated lomax distribution as given in (10) and , and  > 0, 1  r < s  n, we have that

Triple moments
For exponentiated lomax distribution as given in (12)    Substituting I (x) in (18-A), we found that By using the same way that we used to get I (y) and I (x), we obtain (18).

Recurrence relations for single and product moments
In this section, the recurrence relations for the single and product moment of the ELD are established as follows: from (1) and (2)  r-2 f(x)dx for integration and the rest of the integrand for differentiation, we get  Note that, when  = 1,  =  = 1 and  = 1, we obtain the recurrence relations for product moments from lomax distribution, standard pareto distribution and exponentiated pareto distribution respectively.

Percentage Points of order statistics
The cumulative distribution function of x r:n , 1  r  n is given by F r:n (x) = I F(x) (r, n-r+1) Where F(x) = F (x; , , ) is cdf given in (2) and Where I F(x) (r,n-r+1) is the incomplete beta function. Therefore, the l00p th percentile of x r:n for given n,r and p can be obtained by solving the following equation (Raqab 1998).
The percentage points can be calculated from (23) either by using the tables of incomplete beta function prepared by Pearson (1934)  The following tables (2) and (3) gives values of 100 p th percentage points of X 1:n,p and X n:n,p for n = 2, 5, 10, 15 and for P = 0.1 to 0.9 when  = 1.5, 5,  = 1.2, 2 and  = 2, 2.5  When 1<r<n, the percentage points can also be obtained by using t-approximation to in complete beta function (Ojo 1988) as follows: Let T v be a t random variable with v degrees of freedom which is obtained by equatating the coefficient of kurtosis of the generalized logistic to that of the t-distribution. Denote the r th cumulant of the logistic distribution by k r . The approximate expression for the F r:n (x) is found to be Where This immediately given Approximation to percentiles of order statistics x r:n , (1  r  n) can be obtained from (24) by using t-table.

Conclusion
This study deals with the order statistics from the exponentiated Lomax distribution (ELD). Explicit forms for the single, product and triple moments of the order statistics from ELD are derived. Coefficients of skewness and kurtosis are calculated for different values of sample size and distribution parameters. Some recurrence relations for both single and product moments are established. Also, the percentage points of the r th order statistic from ELD are presented and computed for the smallest and the largest order statistics at different values of sample size and distribution parameters.