# Characterisation and Some Statistical Aspects of Univariate and Multivariate Generalised Pareto Distributions

[Doktorsavhandling]

Extreme value theory is about the distributions of very large or very small values in a time series or stochastic process. This has numerous applications connected with environmental science, civil engineering, materials science and insurance. A rather recent approach for modelling extreme events is the so called peak over threshold (POT) method. The generalised Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold.

This thesis consists of three papers. The main focus is on some theoretical and applied statistical issues of univariate and multivariate extreme value modelling. In the first paper we compare the empirical coverage of standard bootstrap and likelihood-based confidence intervals for the parameters and 90\%-quantile of the GPD. By applying a general method of D. N. Lawley, small sample correction factors for likelihood ratio statistics of the parameters and quantiles of the GPD have been calculated. The article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.

In the second paper we give a multivariate analogue of the GPD and consider estimation of parameters in some specific bivariate generalised Pareto distributions (BGPD's). We generalise two of existing bivariate extreme value distributions and study maximum likelihood estimation of parameters in the corresponding BGPD's. The procedure is illustrated with an application to a bivariate series of wind data.

The main interest in the thesis has been on practicality of the methods so when a new method has been developed, it's performance has been studied with the help of both real life data and simulations. In the third paper we use three previous articles as examples to illustrate difficulties which might arise in application of the theory and methods which may be used to solve them. A common theme in these articles is univariate and multivariate generalised Pareto distributions. However, the discussed problems are of a rather general nature and demonstrate some typical tasks in applied statistical research. We also discuss a general approach to design and implementation of statistical computations.

**Nyckelord: **generalised Pareto distribution, multivariate extreme value theory, multivariate Pareto distribution, small sample properties, Bartlett's correction, maximum likelihood, statistical computations, simulation AMS 1991 subject classification: 62F11, 62E20, 60F17, 65U05

Denna post skapades 2006-08-25.

CPL Pubid: 1093