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

Abstract: 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.