AbstractsStatistics

ENGLISH ABSTRACT: Extreme value theory (EVT) involves the development of statistical models and techniques in order to describe and model extreme events. In order to make inferences about extreme quantiles, it is necessary to estimate the extreme value index (EVI). Numerous estimators of the EVI exist in the literature. However, these estimators are only applicable in the single sample setting. The aim of this study is to obtain an improved estimator of the EVI that is applicable to an ANOVA setting. An ANOVA setting lends itself naturally to empirical Bayes (EB) estimators, which are the main estimators under consideration in this study. EB estimators have not received much attention in the literature. The study begins with a literature study, covering the areas of application of EVT, Bayesian theory and EB theory. Different estimation methods of the EVI are discussed, focusing also on possible methods of determining the optimal threshold. Specifically, two adaptive methods of threshold selection are considered. A simulation study is carried out to compare the performance of different estimation methods, applied only in the single sample setting. First order and second order estimation methods are considered. In the case of second order estimation, possible methods of estimating the second order parameter are also explored. With regards to obtaining an estimator that is applicable to an ANOVA setting, a first order EB estimator and a second order EB estimator of the EVI are derived. A case study of five insurance claims portfolios is used to examine whether the two EB estimators improve the accuracy of estimating the EVI, when compared to viewing the portfolios in isolation. The results showed that the first order EB estimator performed better than the Hill estimator. However, the second order EB estimator did not perform better than the “benchmark” second order estimator, namely fitting the perturbed Pareto distribution to all observations above a pre-determined threshold by means of maximum likelihood estimation. AFRIKAANSE OPSOMMING: Ekstreemwaardeteorie (EWT) behels die ontwikkeling van statistiese modelle en tegnieke wat gebruik word om ekstreme gebeurtenisse te beskryf en te modelleer. Ten einde inferensies aangaande ekstreem kwantiele te maak, is dit nodig om die ekstreem waarde indeks (EWI) te beraam. Daar bestaan talle beramers van die EWI in die literatuur. Hierdie beramers is egter slegs van toepassing in die enkele steekproef geval. Die doel van hierdie studie is om ’n meer akkurate beramer van die EWI te verkry wat van toepassing is in ’n ANOVA opset. ’n ANOVA opset leen homself tot die gebruik van empiriese Bayes (EB) beramers, wat die fokus van hierdie studie sal wees. Hierdie beramers is nog nie in literatuur ondersoek nie. Die studie begin met ’n literatuurstudie, wat die areas van toepassing vir EWT, Bayes teorie en EB teorie insluit. Verskillende metodes van EWI beraming word bespreek, insluitend ’n bespreking oor hoe die optimale drempel bepaal kan word. Spesifiek word twee aanpasbare…