Regression analysis for extreme value responses and covariates

Abstract

Extreme value theory concerns with modeling rare and extreme events. It provides tools for making probabilistic statements about occurrence of extremes and predict their magnitude. As such events can be detrimental it is important to understand which factors affect extremes and how.

This thesis pioneers the development of regression-type models for the situation where both the response and the covariate are extreme. In particular, we propose a quantile regression framework that takes into account the fact that the limiting distribution of component maxima is an extreme value copula. The main parameter of interest in the class of proposed models is what we refer to as the regression manifold. It is comprised of quantile regression lines derived based on a conditional distribution which arises from the multivariate extreme value distribution. We learn about the regression manifold from data via a random Bernstein polynomial prior and illustrate the process on several real data applications.

Another contribution of this thesis rests in the development of shrinkage methods for regression-type models for multivariate extremes so to address the problem of variable selection in this setting. We resort to max-linear models which accommodate non-trivial conditional dependence structures, and obtain constraints on the model parameters that allow to distinguish between different levels of complexity of the resulting regression manifold. These conditions are then incorporated into the model to do inference on the regression manifold through the instrumentality of Bayesian regularisation techniques.