A Characterization of Continuous Distributions via Regression on Pairs of Record Values

Abstract

SummaryThe exponential type is characterized in terms of the regression of a (possibly non-linear) function of a record value with its adjacent record values as covariates. Monotone transformations extend this result to more general settings, and these are illustrated with some specific examples.

Characterizations of probability distributions via bivariate regression of record values

“…The following result is an extension of Theorem 2 in Bairamov et al (2005) to regression on a pair of non-adjacent covariates. As we will see in the next section, it follows from Theorem 1 choosing h(x) = x k+r /(k + r)!.…”

“…Remarks. (i) If k = r = 1, then Theorem 1A coincides with Theorem 1 in Bairamov et al (2005) because, using (9) below, the assumption (2) can be written as M 1 (l F , v) = 0. (ii) The statement in Theorem 1A holds true when k = 1 and r ≥ 1 as well (with r M (l F , v) = 0 instead of (5)) and can be proved along the same lines, differentiating with respect to u instead of v.…”

“…Theorem (Bairamov et al (2005)). Suppose F is absolutely continuous with density f , that h is continuous in [l F , r F ] and continuously differentiable in (l F , r F ), and that almost everywhere in this open interval h ′ (x) = M (l F , x).…”

“…The aim of this paper is to extend the results given in Bairamov et al (2005) by studying characterizations of F in terms of the regression on two non-adjacent record values E[ψ(X(n))|X(n − k) = u, X(n + r) = v] (l F < u < v < r F ),…”

“…as well as i M (u, v) and M j (u, v) for the ith and jth partial derivative of M (u, v) with respect to u and v, respectively. Bairamov et al (2005) characterize the exponential distribution as follows.…”