On estimation and goodness-of-fit testing for the Pareto distribution

Abstract

The Pareto distributions are a class of distributions containing several members with varying levels of flexibility. Two of these members, the Pareto Type I and the Lomax distributions, are of particular interest in this thesis. The Pareto Type I distribution is a popular choice of model in many applications where heavy tailed data are modelled. This flexible distribution if often used to model the tails of, for example, observed insurance claims or salaries. The Lomax distribution is a generalisation of the Pareto Type I distribution (and a special case of the Pareto Type II distribution) which was initially proposed as a model for business failure, but has since found a wide range of other applications. We propose several classes of goodness-of-fit tests for the Pareto Type I distribution based on two characterisations. The first characterisation relates to the distribution of order statistics while the second is based on conditional expectation. We derive the asymptotic properties of the tests based on the first of these characterisations. Extensive Monte Carlo studies are included in order to examine the finite sample power performance of the proposed tests and it is found that these tests are competitive, often outperforming the existing tests considered. In the Monte Carlo power studies mentioned above, we employ different estimators to fit the Pareto distribution. The choice of estimation technique is shown to have a surprisingly profound effect on the resulting empirical powers. This prompted research into a comparison between various parameter estimation techniques. The Lomax distribution is a popular variant of the Pareto class and it is frequently fit to observed financial data. We compare the performance of various classical estimation techniques to techniques based on the minimisation of some distance measure. Interestingly, these minimum distance estimators outperform the classical techniques in various settings. The use of the goodness-of-fit tests developed in the thesis are illustrated using practical data. The same holds for the minimum distance estimators considered.