The risk function of the goodness-of-fit tests for tail models

Abstract

This paper contributes to answering a question that is of crucial importance in risk management and extreme value theory: How to select the threshold above which one assumes that the tail of a distribution follows a generalized Pareto distribution. This question has gained increasing attention, particularly in finance institutions, as the recent regulative norms require the assessment of risk at high quantiles. Recent methods answer this question by multiple uses of the standard goodness-of-fit tests. These tests are based on a particular choice of symmetric weighting of the mean square error between the empirical and the fitted tail distributions. Assuming an asymmetric weighting, which rates high quantiles more than small ones, we propose new goodness-of-fit tests and automated threshold selection procedures. We consider a parameterized family of asymmetric weight functions and calculate the corresponding mean square error as a loss function. We then explicitly determine the risk function as the finite sample expected value of the loss function. Finally, the risk function can be used to discuss the question of which symmetric or asymmetric weight function and, thus, which goodness-of-fit test should be used in a new method for determining the threshold value.

Chasing the Tail: A Generalized Pareto Distribution Approach to Estimating Wealth Inequality

“…The approach taken here is to apply a modified version of the Anderson-Darling right-tail-weighted distance (RTAD) measure (see Chernobai et al [2005] and Hoffmann and Börner [2020]) to select a candidate pair of parameters, and then apply of an adaptation of the Cramer von Mises (CVM) test (see Choulakian and Stephens [2001]) to assess the plausibility that the observed data may be described by a GPD with these parameter estimates.…”

Chasing the Tail: A Generalized Pareto Distribution Approach to Estimating Wealth Inequality

“…The approach taken here is to apply a modified version of the Anderson-Darling right-tail-weighted distance (RTAD) measure (see Chernobai et al [2005] and Hoffmann and Börner [2020]) to select a candidate pair of parameters, and then apply of an adaptation of the Cramer von Mises (CVM) test (see Choulakian and Stephens [2001]) to assess the plausibility that the observed data may be described by a…”