Identification of Parameters by the Distribution of a Maximum Random Variable

Abstract

Summary Under what conditions does the distribution of the maximum of a set of independent random variables uniquely determine the distributions of the component random variables? In the univariate case, a sufficient condition is that for every pair of distinct densities in the family of possible densities, one density converges to zero infinitely faster than the other, as x → ∞. Hence, the distribution of max {X i, i = 1, ..., n} when X i has the distribution N(μi, σ2 i), uniquely determines μi, σ2 i, i = 1, ..., n (except for indexing). The identifiability property is proved for bivariate normal distributions for every n when all correlations are positive; each component of the vector of maxima consists of the maximum of that component of the n constituent vectors.