Topp-Leone generalization of the Generalized Pareto distribution and its impact on Extreme value modelling

Abstract

Extreme Value theory (EVT) is a phenomenon used to model rare or extreme events and has been useful in well-known areas such as finance, economics, hydrology, insurance, etc. In this paper, we combine EVT and Bayesian statistics to estimate the extreme value index and other distribution parameters. EVT studies the behavior of the tails of the distribution, while Bayesian statistics allows us to incorporate prior knowledge of the parameters. The interdependence between these two statistical branches allows us to account for uncertainty in parameter and tail estimation. Block maxima and Peaks over Threshold are EVT divisions that are used to model observations. In this paper we use the Peaks over Threshold approach. The generalized Pareto distribution is a Peaks over Threshold distribution. Existing literature studied the generalizations and extensions of the generalized Pareto distribution. These extensions mostly focus on the positive domain of attraction. In this paper we contribute to the study of EVT by considering both the negative and positive domains of attraction. We consider the (Topp and Leone, 1955) generalization for the generalized Pareto distribution. We show, by means of a simulation study, that this distribution can effectively estimate the extreme value index and that it is less sensitive to threshold selection than the normal generalized Pareto distribution.