Extremal Properties, Asymptotic Independence and the Probability of Failure Sets

Specialeforsvar ved Sophie Marie Martinsen

Titel: Extremal Properties, Asymptotic Independence and the Probability of Failure Sets

Abstract: In this thesis, we examine large fire insurance claims by using extreme value theory both in the univariate and multivariate cases. The fire insurance data consists of already paid claims, and expected payments, which have not yet been paid out.
The aim is to examine the probability of large claims and therefore we estimate tail-probabilities. Hence, the main purpose of this thesis is to estimate the tail indices and tail distributions and find out if there is any dependence between the paid claims and the expected payments. This leads us, in the end, to the estimation of probabilities in failure sets.
We introduce the theory in the univariate case where we analyze the paid claims and the expected payments separately. We estimate the tail indices and the quantiles by using three different methods that all led to different estimates. The small number of observations makes the estimates uncertain, especially for the expected payments.
However, all tail indices indicated that the extreme value distribution is the Fréchet distribution for both the paid claims and the expected payments. The estimation of the tail index and the indication of a Fréchet distribution as the extreme value distribution makes it possible to extend the theory to the multidimensional case. Here, the asymptotic dependence between the paid claims and the expected payments is investigated as we are interested in the collective impact on a catastrophic situation. The spectral measure is used as a tool when examining the dependence structure. We transform the data by using three different methods which then allow us to estimate the spectral measure. The histograms for the estimated spectral measure indicate that the paid claims and the expected payments are asymptotically independent, contrary to what we would expect.
Finally, the theory of failure sets is presented. The failure set consists of combinations of the components which result in an extremely large value. We consider a failure set where we have no observations. The probability of future observations in this chosen failure set is estimated by using extrapolation.

Vejleder: Thomas Mikosch
Censor:  Mette Havning