Extremal values, Asymptotic Indepencece and Hidden Regular Variation

Specialeforsvar ved Jenny Hedermann Overgaard

Titel: Extremal Values, Asymptotic Independence and Hidden Regular Variation

Abstract: In this thesis, we examine large claims for motor insurance data using extreme value theory. The motor insurance data consists of claims that can have up to three different damages. The main purpose is to examine the damages separately and investigate the dependence for the different damages in pairs. \\ We introduce the theory for both the univariate case and the multivariate case that is used for the analysis of the claims. The univariate case is introduced to obtain different methods for estimating the tail index and the tail distribution which also gives the possibility to extend to the multivariate case. The multivariate case allows for an investigation of asymptotic dependence for the damages. The angular measure, which is also known as the spectral measure, turns out to be important when we examine asymptotic dependence. Theory of hidden regular variation is also presented which can be used if the components in data turn out to be asymptotically independent. \\ In the univariate case, we apply three methods for estimating the tail index for each of the three different kinds of damages and the total claim amount. The estimates turn out to be very different for the different methods, and the analyses do not give a definitive answer as to which tail indices and tail distributions to use. The asymptotic dependence of the damages is investigated in pairs. The histograms for the estimated angular measure strongly indicates that the pairs are asymptotically independent. Since the pairs are asymptotically independent, we estimate the hidden angular measures. The hidden angular measures estimated on a compact subset do not show any sign of the pairs being strongly asymptotically dependent, however, the clear signs of asymptotic independence of the pairs have vanished.

Vejleder: Thomas V. Mikosch
Censor:   Mette M. Havning