Some characterizations of discrete distributions based on linear regressions of adjacent order statistics (Q1112486)

Let \(X_{(1)}\leq X_{(2)}\) denote the order statistics of a random sample of size 2 from a discrete distribution taking integer values. We determine classes of such distributions for which the best predictors of \(X_{(i)}\), given \(X_{(j)}\) and the event \(\{X_{(2)}-X_{(1)}\geq m\}\), namely \(E(X_{(i)}| X_{(j)},X_{(2)}-X_{(1)}\geq m)\), is a linear function of \(X_{(j)}\), for \(i\neq j\) where m is a nonnegative integer. This results in new characterizations of the geometric and discrete uniform distributions. The discussion is extended to a larger sample size but is limited to regressions of adjacent order statistics.