Extreme Value Analysis and Failure Set Probabilities - An Application to the Danish Fire Insurance data

Specialeforsvar ved Tinna Sommer Nielsen og Zenia Katrine Hansen

Titel: Extreme Value Analysis and Failure Set Probabilities - An Application to the Danish Fire Insurance Data  

Abstract: In this thesis we will be modelling extremal events through the Danish Fire Insurance data. The data consists of fire losses collected from1980-1990, with different components; Building, Content, Loss of Profit and an assembled one called Total. The focus on extreme events is very important because of its danger to the insurance company, so an analysis can therefore be used to manage this risk. Initially we introduce the univariate theory with the purpose of estimating the tail index and the marginal tail distributions in different ways. This gives the opportunity to extend to the multivariate case where the asymptotic dependence structure for different component pairs are analysed. The spectral measure contributes to the investigation of the asymptotic dependence structure, and for that the data has to be tail equivalent hence data transfor-mation is introduced. At last the theory for estimating failure set probabilities for different quantiles is presented, where a method from the univariate analysis is revisited. In the univariate analysis five methods are used for tail estimation of the different components. In all cases we ended up with a positive ξ, stating that we are in the maximum domain of attraction of the Fréchet case i.e. heavy tailed distributions. To extend to the multivariate analysis, data is transformed to standard case, using the Power and Rank method. The spectral measures are being estimated and histograms of probability densities are being plotted which shows asymptotic independence for all component pairs. This is a surprising discovery, since asymptotic dependence is expected



Vejleder:   Thomas Mikosch
Censor:     Mette M. Havning