On statistical model extensions based on randomly stopped extremes

TL;DR

This paper introduces new statistical model transformation mechanisms based on randomly stopped extremes, offering alternatives to traditional extreme value models that rely on large sample theory, and explores their properties and stability.

Contribution

It develops novel model extension methods using randomly stopped maxima and minima, expanding the toolkit for analyzing extreme value data beyond asymptotic approaches.

Findings

Identifies conditions under which model extensions are stable.

Characterizes stopping models that produce equivalent extensions for maxima and minima.

Demonstrates advantages of these models through practical examples.

Abstract

The maxima and the minima of a randomly stopped sample of a random variable, $X$, together with two newly defined random variables that make $X$ into the maxima or minima of a randomly stopped sample of them, can be used to define statistical model transformation mechanisms. These transformations can be used to define models for extreme value data that are not grounded on large sample theory. The relationship between the stopping model and characteristics of the corresponding model transformations obtained is investigated. In particular, one looks into which stopping models make these model transformations into model extensions, and which stopping models lead to statistically stable extensions in the sense that using the model extension a second time leaves the extended model unchanged. The stopping models under which the extensions based on randomly stopped maxima and their inverses coincide with the extensions based on randomly stopped minima and their inverses are also characterized. The advantages of using models obtained through these model extension mechanisms instead of resorting to extreme value models grounded on asymptotic arguments is illustrated by way of examples.