A characterization of the Gumbel distribution based on record values

Abstract

Suppose {X-j, j = 1,2,...} be a sequence of independent and identically distributed random variables from Type I extreme value distribution. Let X-L(1), X-L(2),... be the lower record values of this distribution. We assume the first observation as the first lower record, X-L(1). It is known that X-L(m) is identically distributed as XL(m+1) + W/m, where W has the exponential distribution with cumulative distribution function F(x) = 1-e(-x), x>0. In this article we will characterize the Type I extreme value distribution using this distribution property.