Second Order Von Mises’ Condition For Maximum Domain Of Attraction

Extreme value theory is a separate branch of statistics that is used to analysis the statistical regularity of extreme events by the methods. It is widely applied in natural science and social science. This theory is based on the Fisher-Tippett Theory, stating that there are three types of distributions for extreme value distributions. How to judge whether a distribution function belongs to a maximum domain of attraction is the important content of extreme value theory.This paper mainly includes three parts. The research process of EVT and some lemmas that will be used later are introduced in chapter one. In the second chapter, this paper firstly discusses five equivalent conditions for maximum domain of attraction, and then discusses the necessary and sufficient conditions of distribution functions belonging to Gumbel distribution,Frechet distribution and Weibull distribution.In1936, Von Mises provided a series of sufficient conditions to val-idate distribution functions belonging to the maximum domain of at-traction. Inspired by these, this paper gives the Von Mises’conditions when the distribution has first or second order derivative in chapter3. At last, three common continuous distributions are used to verify that they respectively belong to three kinds of extreme value distribution.