A mixture model approach to extreme value analysis of heavy tailed processes

Abstract

Extreme value theory (EVT) encompasses statistical tools for modelling extreme events, which are defined in the peaks-over-threshold methodology as excesses over a certain high threshold. The estimation of this threshold is a crucial problem and an ongoing area of research in EVT.

This dissertation investigates extreme value mixture models which bypass threshold selection. In particular, we focus on the Extended Generalised Pareto Distribution (EGPD). This is a model for the full range of data characterised by the presence of extreme values. We consider the non-parametric EGPD based on a Bernstein polynomial approximation. The ability of the EGPD to estimate the extreme value index (EVI) is investigated for distributions in the Frechet, Gumbel and Weibull domains through a simulation study. Model performance is measured in terms of bias and mean squared error. We also carry out a case study on rainfall data to illustrate how the EGPD fits as a distribution for the full range of data. The case study also includes quantile estimation.

We further propose substituting the Pareto distribution, in place of the GPD, as the tail model of the EGPD in the case of heavy-tailed data. We give the mathematical background of this new model and show that it is a member of the EGPD family and is thus in compliance with EVT. We compare this new model's bias and mean squared error in EVI estimation to the old EGPD through a simulation study. Furthermore, the simulation study is extended to include other estimators for Frechet-type data. Moreover, a case study is carried out on the Belgian Secura Re data.