Regression Models for Extreme Values with Random Function Covariates

Abstract

A fundamental principle of statistics of extremes is that any realistic quantification of risk re quires extrapolating into a distribution’s tail—often beyond the observed extremes in a dataset. Yet, as modern technology advances, an increasing amount of data is recorded continuously or intermittently, and hence the question arises: how to take advantage of such data in an extreme value framework? Motivated by this question, this thesis develops a class of novel statistical methods that can be used for marginal and joint distributions to learn how the extreme values may change according to a functional covariate. The first contribution consists of a functional regression model for the tail index that can be used for assessing how the magnitude of the extremes can change according to a random function. Another contribution of this thesis is the development of a nonparametric regression model that can be regarded as a functional covariate regression method, designed for situations where there is a need to assess how the extremal dependence of a random vector can change according to a functional explanatory variable. Such development is based on modeling a family of angular measures indexed by a random function. The performance of the proposed methodologies is assessed via numerical studies, and financial data is used to illustrate their application