Extreme Value Analysis of Bivariate Heavy Tailed Insurance data

Specialeforsvar ved Siri Louise Weicker

Titel: Extreme Value Analysis of Bivariate Heavy Tailed Insurance Data

Abstract: The characteristics of an extreme event from an (re)insurance point of view is, that it is a rare event with a large financial impact. Therefore among other things, good estimates of the tail of the large claims’ distributions are necessary for determining expected future losses, premiums etc. For this, extreme value theory, EVT, is needed, which includes statistical techniques when analyzing data with only a few observations. This thesis examines the size of large claims of a bivariate insurance data set consisting of general liability claims. The aim of this thesis is to gather theoretical results from univariate and bivariate EVT and illustrate their application. In the univariate case, the variables are ana-lyzed separately for estimating the tail index and the tail distribution. The analysis is focused on the generalized Pareto Distribution, GPD, where different methods are used to estimate its parameter(s). In the bivariate case, the dependence structure between the two types of claims is investigated. Particular attention is drawn to estimate the probability that a pair of random variables will take values in some given extreme set - also referred to as a failure set. An essential tool for doing this is the spectral (or angular) measure. The univariate analysis evaluates three different methods used for fitting a GPD to the excesses above some chosen high threshold. The three methods give different results and the Peaks over Threshold, POT, method approximates the behaviour of the observations as well as high quantiles best. Turning to the bivariate analysis, the estimated spectral measure does, among the extremes, indicate asymptotic dependence, although this is not crystal clear. Last, but not least, using the spectral measure for estimating the probability of an event to fall in some chosen remote failure set, is indeed favorable - especially when no claims have been observed in that area yet.

Vejleder: Thomas Mikosch
Censor:   Yuri Goegebeur, SDU