Characterization of Distributions by Using the Conditional Expectations of Generalized Order Statistics

Abstract

In this study, some continuous distributions through the proper- ties of conditional expectations of generalized order statistics are characterized. Let $X_{1:n:m:k},...,X_{n:n:m:k}$ be the generalized order statistics, where n $nin Bbb{N},k>0;m_1,...,m_{n-1}in Bbb{R},M_r = sumlimits_{j=r}^{n-1}mj,1leq rleq n-1,gamma_r= k+n-r+M_r > 0$ for all $rin {1,...,n-1}$ and let m = ${m_1,...,m_{n-1}}$, if n =arbitrary, if n = 1. Characterization theorems for a general class of distributions are presented in terms of the function $E {g(X_{j:n:m:m+1)| X_{j-p:n:m:m+1} = X,_{j+q:n:m:m+1} = y}$ = A(x; y); where k = m+1, p and q are positive integers such that $p+1leq jleq n-q$ and g(.), A(., .) is a real valued function satisfying certain regularity conditions.