Identification of Parameters by the Distribution of a Maximum Random Variable.

摘要

The distribution of the maximum of a set of independent random variables uniquely determine the distributions of the component random variables under certain conditions. In the univariate case a sufficient condition is roughly that for every two distinct densities f(x) and g(x) in the family of possible densities f(x)/g(x) approaches 0 or infinity as x approaches infinity. Hence, the distribution of max X(i), i = 1, ..., n, when X(i) has the distribution N mu sub i, sigma squared sub i uniquely determines mu sub i, squared sub i, i=1, ..., n (except for indexing). The identifiability property has been proved for multivariate normal distributions for n = 2 and 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. Inequalities for the probability in the upper right-hand quadrant of the bivariate normal distribution have been developed; these are generalizations of Mills' ratio.