Some Characterization Results Based on Conditional Expectation of Function of Dual Generalized Order Statistics

Two families of probability distributions are characterized through the conditional expectations of dual generalized order statistics ( ), conditioned on a non-adjacent dual generalized order statistics. Also a result based on the unconditional expectation and a conditional expectation of is used to characterize family of distributions. Further, some of its deductions are also discussed.


Introduction
The concept of generalized order statistics ( gos ) has been introduced as a unified approach to a variety of models of ordered random variables with different interpretation (Kamps, 1995), such as ordinary order statistics, sequential order statistics, progressive type II censoring, record values and Pfeifer's records.Generalized order statistics serve as a common approach to a structural similarities and analogies.Several of these models can be effectively applied, e.g., in reliability theory.Using this concept of gos , Burkschat et al. (2003) introduced the concept of dual generalized order statistics ( dgos ) that enables a common approach to descending ordered random variables like reverse ordered order statistics, lower record values etc.


be a sequence of independent and identically distributed ) (iid random variable with absolutely continuous distribution function and the probability density function are called dgos if their joint pdf is given by Here we will assume two cases:  . The pdf of the th r dgos is given by (Burkschat et al., 2003) The joint pdf of the th r and th s dgos is where Case II: The pdf of the th r dgos is given by (Burkschat et al., 2003) and the joint pdf of the th r and th s dgos is and order statistics, from a sample of size n and when reduces to the th r  k lower record value (Pawlas and Szynal, 2001).A number of results on characterization of distributions of dual generalized order statistics are available in the literature.For a detailed survey one may refer to Ahsanullah (2004), Mbah and Ahsanullah (2007)

Characterizations of distributions when
Theorem 2.1: Let X be an absolutely continuous random variable with the df , where  and  may be finite or infinite, then for is a monotonic and differentiable function of .


, in view of (1.9) and (1.12)This proves the necessary part.
For the sufficiency part, we have at Rearranging the terms in (2.4) and noting that 0 , we have Using the result (Khan et al., 2006) x h a and hence the Theorem.Khan et al. (2010).

Remark 2.2:
, we will get the following result for order statistics for  in equation (2.9), we will get the result for order statistics as follows, Theorem 2.2: Let X be an absolutely continuous random variable with the df , where  and  may be finite or infinite, then for (2.12) where Proof: First we shall prove that (2.12) implies (2.11).In view of Khan et al. (2010) This proves the necessary part.

Remark 2.5:
, we will get result for order statistics as follows,

Remark 2.7: At
as obtained in Theorem 2.1.

Conclusion
In this paper, conditional expectation of the difference of two dgos conditioned on nonadjacent dgos are considered to characterize the df .13) Khan et al. (2010) have characterized the general class of distributions through conditional expectation of dgos conditioned on non-adjacent dgos .We have extended the result of Khan et al. (2010) for the difference of the conditional expectations conditioned on non-adjacent dgos , its particular cases for order statistics, lower record statistics as obtained by Khan et al. (2011) and Faizan and Khan (2011) are discussed.