Asymptotic Distribution of Extremes.

Abstract

Abstract : This final report summarizes the results obtained by the principal investigator on the asymptotic distribution of extremes during the years 1978-1981. These results discuss the role of the choice of the population distribution in a classical model for extremes (including characterizations) as well as the necessity for deviating from the classical model for describing real life situations. Several classes of dependent random variables, with considerable detail on exchangeability, are discussed with the main focus on the behavior of their extremes. Particular emphasis is put on reliability applications and on material strength distributions. Because of the prominent role of the exponential distribution in reliability theory, the principal investigator collected material on this distribution (both on the applicability of the exponential distribution in model building and a large variety of techniques in statistical inference). In addition to asymptotic distributions, sharp inequalities are discussed which can be applied with success to two types of problems of extreme value theory: estimating system reliability and determining whether a particular assumed model contradicts reality. Extensions of some univariate results on domains of attraction to a particular asymptotic extreme value distribution are given to the multivariate case. In the univariate case, a statistical test is proposed on deciding whether the population distribution belongs to the domain of attraction of a particular extreme value distribution. (Author)