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

Let X1, X2, . . . be independent copies of a random variable X whose distribution function is denoted by F. There have been many studies on characterizations of F via linear regression relations of one order statistic on one or two other order statistics. See Wesolowski & Ahsanullah (1997) for references and the most complete results. Similarly, the regression of one record value on another has received attention. Denote record times by L(1) = 1 and, for n > 1, L(n) = min{j : j > L(n− 1) and Xj > XL(n−1)} , and corresponding record values by X(n) = XL(n) ; see Nevzorov (2001). Fix positive integers n and s, and real constants a and b. The one-variable linear regression problem is to determine all F for which