Tail Index Estimation of the Generalised Pareto Distribution using a Pivot from a Transformed Pareto Distribution

Abstract

In extreme value analysis, the Generalised Pareto (GP) is an important statistical distribution for modelling tails of several phenomena. The tail index for this distribution plays a vital role as it determines the tail heaviness of the underlying distribution and it is the primary parameter required for the estimation of other extreme events. The estimation of the tail index of the GP distribution is addressed in this paper. The standard methods, such as maximum likelihood and probability weighted moments, are known to perform badly in small samples and to provide estimates that are inconsistent with observed values respectively. In this paper, the parameters of the GP distribution are estimated using a transformation to the Pareto distribution. Unlike the GP distribution, explicit expressions exist for the maximum likelihood estimators of the parameters of the Pareto distribution. In addition, a linear transformation of the distribution function enables the estimation of the tail index independent of the scale parameter. The proposed estimators are compared with the maximum likelihood estimator through a simulation study. The results show that the performance of the estimators was better, and at worst, approximately equal in performance to the standard method. We illustrate the application of the estimators with real data on insurance claims.