'n Ondersoek na die eindige steekproefgedrag van inferensiemetodes in ekstreemwaarde-teorie
Thesis (MComm (Statistics and Actuarial Science))--University of Stellenbosch, 2005.
Extremes are unusual or rare events. However, when such events – for example earthquakes, tidal waves and market crashes - do take place, they typically cause enormous losses, both in terms of human lives and monetary value. For this reason, it is of critical importance to accurately model extremal events. Extreme value theory entails the development of statistical models and techniques in order to describe and model such rare observations. In this document we discuss aspects of extreme value theory. This theory consists of two approaches: The classical maxima method, based on the properties of the maximum of a sample and the more popular threshold theory, based upon the properties of exceedances of a specified threshold value. This document provides the practitioner with the theoretical and practical tools for both these approaches. This will enable him/her to perform extreme value analyses with confidence. Extreme value theory – for both approaches - is based upon asymptotic arguments. For finite samples, the limiting result for the sample maximum holds approximately only. Similarly, for finite choices of the threshold, the limiting distribution for exceedances of that threshold holds only approximately. In this document we investigate the quality of extreme value based inferences with regard to the unknown underlying distribution when the sample size or threshold is finite. Estimation of extreme tail quantiles of the underlying distribution, as well as the calculation of confidence intervals, are typically the most important objectives of an extreme analysis. For that reason, we evaluate the accuracy of extreme based inferences in terms of these estimates. This investigation was carried out using a simulation study, performed with the software package S-Plus.