Characterizations of distributions via linearity of regression of generalized order statistics

Abstract

Let X (r, n, m˜, k), 1 ≤ r ≤ n, denote generalized order statistics based on an absolutely continuous distribution function F. We characterize all distribution functions F for which the following linearity of regression holds

E(X(r+l,n,m˜,k) | X(r,n,m˜,k))=aX(r,n,m˜,k)+b.

We show that only exponential, Pareto and power distributions satisfy this equation. Using this result one can obtain characterizations of exponential, Pareto and power distributions in terms of sequential order statistics, Pfeifer’s records and progressive type II censored order statistics.