Relations for Single and Product Moments of Odds Generalized Exponential-Pareto Distribution Based on Generalized Order Statistics and Characterization

This paper deals with explicit expressions and recurrence relations for single, inverse, product and ratio moments of Generalized Order Statistics from Odds Generalized Exponential-Pareto Distribution (OGEPD). Characterization results have also been carried out.


Introduction
The concept of generalized order statistics (gos) was first introduced by [8] which envelops variety of models of ordered random variables that acts as a flexible model in various directions such as order statistics, upper record values, progressive type II censoring order statistics, sequential order statistics and Pfeiffer's records.Let X 1 , X 2 , ... be a sequence of independent identically distributed (iid) random variables with distribution function (df ) F (x) and probability density function (pdf ∑ n−1 j=r m j such that γ r = k + n − r + M r > 0 ∀ r ∈ {1, 2, ..., n − 1}.Then X (r, n, m, k), r = 1, 2, . . ., n, is called gos based on F (x), if their joint pdf is of the form For different values of m i 's, k and γ i 's the model of gos reduces to various models e.g., when (m 1 = m 2 = ... = m n−1 = 0, k = 1, γ i = 1 + n − i), this model reduces to order statistics and for k th upper record values (m 1 = m 2 = ... = m n−1 = −1, i.e. γ i = k, k ∈ N ).Here we take the case m i = m j = m.Then the density function of r th gos X (r, n, m, k) is given by 161 The joint pdf of X (r, n, m, k) and X (s, n, m, k), where A lot of work has been carried out by several researchers in the field of gos.Recurrence relations for moments of gos for specific as well as for general class of distribution have been well investigated hitherto.[2] and [11] derived the recurrence relations for single and product moments of gos for some general class of distributions, whereas explicit expressions for exact moments of gos were probed by [4] and the generalized record values from additive Weibull distribution derived by [13].For more details of gos we have [9,12,3,1,14,10] and [6].
A random variable X is said to have OGEPD [15] if its df is given by The corresponding pdf is of the form The relation between F (x) and f (x) can be easily derived as where

Explicit expressions for single moments
We first derive explicit expression for single moments of r th gos, X (r, n, m, k).The following theorem shows the explicit expression for E and for m = −1, we have 162 RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF ODDS GENERALIZED EXPONENTIAL-PARETO

Special Cases
(i) For m = 0, k = 1 in (6), we get the explicit formula for the single moments of ordinary order statistics for the OGEPD as (ii) For k = 1 in (7), we get the explicit expression for the single moments of upper record values for the OGEPD as is satisfied.

Proof
To obtain the recurrence relation for single moments of gos, we use the result of [2] for the OGEPD Using ( 5) in ( 13), we get On simplifying the above expression, we get the required result of (12).

Special Cases
(i) For m = 0, k = 1 in (12), we get the recurrence relations for single moments of ordinary order statistics for the OGEPD as , we get the recurrence relations for single moments of upper record values for the OGEPD as

Relations for inverse moments
In this section, we derive recurrence relation for inverse moments of gos.The inverse moments of gos are defined as Theorem 3 Fix a positive integer k and for df (3) with 2 ≤ r ≤ n, j = 0, 1, 2, ... the following recurrence relation is satisfied.

Proof
From ( 14) and ( 1), we have Proceeding in similar manner as we have done for Theorem 2, we get ] γr g r−1 m (F (x)) dx.
By using (5) in above expression, we have On simplifying (16), we get the required result of (15).

Special Cases
(i) For m = 0, k = 1 in (15), we get the recurrence relation for inverse moments of ordinary order statistics for the OGEPD as 15), we get the recurrence relation for inverse moments of upper record values for the OGEPD as

Theorem 4
For the OGEPD given in (3), is satisfied.

Proof
From (2), we have By considering the result of [2] for product moments of any distribution, we have Using the relation ( 5) in ( 18) for the OGEPD, we have On simplifying (19), we get the result of (17).

Special Cases
(i) For m = 0, k = 1 in (17), we get the recurrence relations for product moments of ordinary order statistics for the OGEPD as (ii) For m = −1, k = 1 in (17), we get the recurrence relations for product moments of upper record values for the OGEPD as

Relations for ratio moments
The ratio moments of gos are defined as Theorem 5 For the OGEPD given in (3), 1 ≤ r < s ≤ n − 1, k ≥ 1 and j ≥ 0 the following recurrence relation for ratio moments is satisfied.

Proof
We derive the ratio moments for the OGEPD by using (20) and proceeding in similar way as Theorem 4, we have ] γs dydx.
On using (5), we get the result of (21) .

Special cases
(i) For m = 0, k = 1 in (21), we get the recurrence relations for ratio moments of ordinary order statistics for the OGEPD as r,s:n .
(ii) For m = −1, k = 1 in (21), we get the recurrence relations for ratio moments of upper record values for the OGEPD as 6. Characterization

Theorem 6
For the positive integers k and j, a necessary and sufficient condition for a random variable X to be distributed with cdf given in ( 3) is that

Proof
The necessary part follows immediately from (12).On the other hand if the recurrence relation in ( 22) is satisfied, then from (1), we have ] γr .
Differentiating ξ (x) with respect to x, we have Thus, we have Integrating left hand side of (24) and using the expression of ξ (x), we have Using generalization result of Müntz-Szász Theorem of [7], we get F (x) f (x) = a θ λθ x 1−θ which proves that, By simplifying above expression, we can also get the following result, , x > a, θ, λ > 0.

Theorem 7
If X is an absolutely continuous positive random variable with df G (x)and g (x) such that E (X) exist, then holds if and only if , x ≥ a, θ, λ > 0.

Proof
We first prove the necessity part.For this, we consider then, we take On simplifying (26) and using g(x), we get du.
Differentiating above equation with respect to x, we have
Hence, using the result of Lemma 6.1 from [16], we have ) .

Theorem 8
Suppose that X is an absolutely continuous random variable with cdf G (x) such that G (a) = 0 and G (x) > 0 for all x > a.We assume that the pdf of X and g (x) and g ′ (x) exists for all x > 0 and E (X) also exists.Then , x ≥ a, θ, λ > 0.

Proof
We have du.
By differentiating the above expression, we have Using result of Lemma 6.2 from [16], we have Integrating both sides with respect to x, we have where c is a constant determined by using the condition ∫ ∞ a g (x) dx = 1, hence, we get , x ≥ a, θ, λ > 0, which implies (4).

Conclusion
In this paper, exact form for single moments from OGEPD has been established in conjunction with derivations for some recurrence relations for single, inverse, product and ratio moments.Further, first four moments of gos and upper record values from the said distribution have been calculated.A characterization by two methods is also given.

Table 1 (
b): First Four Moments of Upper Record Values