Componentwise Maxima and Threshold Excess Multivariate Extremes: Mutual Connections, their Simple Extreme Distributions and Random Generation Methods for those Extreme Values

Abstract

Multivariate extreme value theories are reviewed in the point of view of the intensity of Poisson process of generating the extreme values. Componentwise maxima (CM) have been principally studied in the conventional researches while the multivariate Pareto distribution was derived a quarter century ago to deal with threshold excess multivariate extremes (TEXMEX). It will be found that the essential concepts and perspectives are not yet complete to comprehend the properties of the multivariate extreme value in the practical sense. Thus the mutual connections between CM and TEXMEX are discussed, and it is especially shown not only CM but also TEXMEX employ their own simple distributions to manifest a method of generating their random values simultaneously.