On characterizations of distributions by regression of adjacent generalized order statistics

Abstract

Let ${{X_{*}^{(r)}}}$ , r ≥  1, denote generalized order statistics, with arbitrary parameters ${\gamma_{1},\dots,\gamma_{r}}$ , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. ${E\big( \psi\big(X_{*}^{(r)}\big) | X_{*}^{(r+1)} \big)=g\big(X_{*}^{(r+1)}\big)}$ where ${\psi,g:\mathbb{R}\mapsto\mathbb{R}}$ are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some ${c,\,d\in\mathbb{R}}$.