On testing the assumption of the Pareto type I distribution

Abstract

The Pareto distribution is a popular model in economics, finance and actuarial sciences. Consequently, a number of goodness-of-fit tests, based on diverse characteristics, have been developed to test the goodness- of-fit hypothesis that an observed data set is realised from this distribution. This thesis reviews the ex- isting literature on goodness-of-fit tests for the Pareto distribution and compares the power performances of these tests. Additionally, several new classes of tests for the Pareto distribution, based on various characterisations, are developed. The first characterisation used for the development of the tests relates to the distribution of the sample minimum. We estimate the characteristic function of the sample minimum using U and V statistics and we propose two classes of tests based on these statistics. The resulting tests are shown to be consistent against fixed alternatives and the finite sample power performances of these tests are demonstrated to be competitive against those of the existing tests in the literature. A second characterisation used is a multiplicative version of the memoryless property. We base two goodness-of-fit tests on this property and we demonstrate that these tests perform well using a finite sample power study. We also consider the setting in which random right censoring is present. We propose a new fixed point characterisation based test for the Pareto distribution in this setting. The characterisation is based on Stein’s method for the approximation of integrals. The power performance of the proposed test is considered against a range of alternative distributions using various censoring proportions. The Pareto distribution has a shape parameter, β > 0, which is typically required to be estimated in order to perform goodness-of-fit testing. A result of independent interest considered is the effect of the estimation method used on the powers achieved by various goodness-of-fit tests. That is, we consider the use of both maximum likelihood and method of moments estimators. It is found that, not only does the estimation technique used have a profound effect on the numerical powers achieved by the various tests, but it also influences the way in which critical values are approximated. Specifically, when maximum likelihood is used, the critical values are shape invariant and not a function of the estimated value of β, meaning that a fixed critical value can be obtained using Monte Carlo simulation. On the other hand, if the method of moments is used, then the critical values are required to be estimated using a parametricbootstrap procedure.