Dispersion Models for Extremes

TL;DR

This paper develops a new class of models called extreme value dispersion models, extending exponential family concepts to better understand the behavior of extreme events and their convergence properties.

Contribution

It introduces the slope function as an analogue to the variance function and characterizes well-known extreme value distributions through quadratic and power slope functions.

Findings

Characterization of extreme value families via slope functions.

Convergence theorem linking slope functions to classical extreme value limits.

Unified framework for analyzing extremes using dispersion models.

Abstract

We propose extreme value analogues of natural exponential families and exponential dispersion models, and introduce the slope function as an analogue of the variance function. The set of quadratic and power slope functions characterize well-known families such as the Rayleigh, Gumbel, power, Pareto, logistic, negative exponential, Weibull and Fr\'echet. We show a convergence theorem for slope functions, by which we may express the classical extreme value convergence results in terms of asymptotics for extreme dispersion models. The main idea is to explore the parallels between location families and natural exponential families, and between the convolution and minimum operations.