Expectation Properties of Generalized Order Statistics from Poisson Lomax Distribution

The Poisson Lomax distribution was proposed by [3], as a useful model for analyzing lifetime data. In this paper, we have derived recurrence relations for single and product moments of generalized order statistics for this distribution. Further, characterization of the distribution is carried out. Some deductions and particular cases are also discussed.


Introduction
The Poisson Lomax distribution was proposed by [3], as a three-parameter lifetime distribution with upside-down bathtub shaped failure rate and heavy tailed, and can be used in modelling many practical situation. It is a compound distribution of the zero-truncated Poisson and the Lomax distributions.
A random variable X is said to follow the Poisson Lomax distribution (PLD) if its probability density function (pdf ) is of the form and the corresponding survival function is whereF (x) = 1 − F (x).

EXPECTATION PROPERTIES OF GENERALIZED ORDER STATISTICS
Relation (3) can also be expressed as where [α(1 − a) + 1)] is an integer.

Generalized order statistics
The concept of generalized order statistics was introduced and extensively studied by [17], which includes different ordered random schemes, such as order statistics, record values, sequential order statistics, progressively type II censored order statistics and Pfeifer's records as its special cases.
Let n ≥ 2 be a given integer andm = (m 1 , m 2 , . . . , m n−1 ) ∈ R n−1 , k ≥ 1 be the parameters, such that The random variables X(1, n,m, k), X (2, n,m, k), . . . , X(n, n,m, k) are said to be generalized order statistics from an absolutely continuous distribution function F () with the probability density function (pdf ) f (), if their joint pdf is of the form If m i = 0; i = 1 . . . n − 1, k = 1, we obtain the joint pdf of the order statistics and for m i → −1, k ∈ N , we get joint pdf of k th upper record values.
Here we may consider two cases: In view of (5), the pdf of r th gos X(r, n,m, k) is given as ( [18]) where The joint pdf of X(r, n,m, k) and X(s, n,m, k), 1 ≤ r < s ≤ n, is given as ( [18]) Case II : m i = m, i = 1, 2, . . . , n − 1.
A large volume of work has been done on the study of moments and recurrence relations between moments of generalized order statistics. The moments of ordered random schemes assume considerable importance in the statistical literature. Many authors have investigated and derived several recurrence relations and identities satisfied by the single as well as product moments. [24], [25] studied the recurrence relations and identities for moments of order statistics for some specific distributions. Recurrence relations for the expected values of certain functions of order statistics are considered by [1], [2]. [7] investigated the relations between expected values of functions of gos. For more detailed survey, one may refer to [4], [5], [8], [12], [15], [18], [19], [26], [27], [29], [30], [32] and references therein.
The characterization of a probability distribution has always been the important topic in statistics and mathematical sciences. Several approaches are available to characterize a probability distribution. In this paper, first we established recurrence relations between single and product moments of gos from PLD. Then, these relation are used to characterize the said distribution. Also a characterization theorem based on conditional expectation is presented. For related results on characterization, one can see [6], [9], [10], [11], [13], [20], [22] and [28] among others.
where B(x, y) is complete beta function.

Proof
We have Now, by using the result from [14] p. 315 given as, we get Hence the theorem.
Theorem2 For the conditions as stated in Theorem 1. The recurrence relation for single moments of gos for PLD is given as Also, Proof We have by [7].
For ξ(x) = x p in (19), recurrence relation for single moments of gos is Now on simplification of (20) and using (14) we get the required result (17).

Remark2
When m i → −1, i = 1, 2, . . . , n − 1, the recurrence relation for single moments of k th upper record values will be

Theorem3
Let case-I be satisfied For the Poisson Lomax distribution given as in (1) and for n ∈ N,m ∈ R, k > 0, 1 ≤ r < s ≤ n, p, q = 1, 2, . . . , then the (p, q)th product moment is given by Consider Since we have from [14], p. 315.
Now using (27) in (24), we get Again consider the integral of (28) as

EXPECTATION PROPERTIES OF GENERALIZED ORDER STATISTICS
Now using (29) in (28), we get the required result.
Under the condition as stated in Theorem 3 The recurrence relation for product moments is given as s, n,m, k)]. (31) Proof We have by [7],
The relation (31) can be proved in view of [7] and using (4).

Characterizations
This section contains characterization results for the given distribution through recurrence relations for single and product moments of gos as well as through conditional moments.
Theorem 5 Fix a positive integer k and let p be a non-negative integer. A necessary and sufficient condition for a random variable X to be distributed with pdf given by (1) is that Proof The necessary part follows from (22). On the other hand, if the relation in (32) is satisfied, then on using [7], for Applying the extension of Müntz-Szász theorem (see, for example, [16]) to (33), we get which proves the theorem.
Theorem 6 Fix a positive integer k and let p and q be non-negative integers. A necessary and sufficient condition for a random variable X to be distributed with pdf given by (1) is Proof The necessary part follows from (31). Now, suppose that the relation in (34) is satisfied. Then, using [7], for ξ(x, y) = x p y q , we have Now applying the extension of Müntz-Szász theorem (see, for example, [16]) to (35), we get which proves the theorem.
Theorem 7 Let X(r, n, m, k), r = 1, 2, . . . , n be the the r th gos based on continuous df F () and E(X) exists. Then for two consecutive values r and r + 1, such that 1 ≤ r < r + 1 ≤ n, if and only ifF Proof [21] have shown that if and only ifF  ) and Comparing (37) with (39), we get Thus, the theorem can be proved in view of (38).
Corollary 5 For the r th order statistics X r:n , r = 1, 2, ...n and under the condition as stated under Theorem 4.3 and consequently if and only ifF It may be noted that similar characterization result can also be seen for adjacent records as

Conclusion
The moments of ordered random variables and recurrence relations between them have received great attention in the past few years in statistical literature. We have obtained exact expressions and recurrence relations for single and product moments of generalized order statistics based on Poisson Lomax distribution. Since generalized order statistics is unified approach for several ordered random variables, thus results obtained can be easily deduced for order statistics, record values, sequential order statistics etc. Characterization theorems that use the properties of sample moments, order statistics, record statistics, and reliability properties can be applied to uniquely determine the associated stochastic model.

Acknowledgement
The authors of this manuscript are thankful to the learned referees for giving their fruitful suggestions in improving this script. The authors are also thankful to editor SOIC for giving their valuable time.