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

TL;DR

This paper develops new goodness-of-fit tests with asymmetric weighting for selecting thresholds in tail modeling, explicitly calculating the risk function to improve risk assessment in extreme value analysis.

Contribution

It introduces a family of asymmetric goodness-of-fit tests and derives their risk functions, enhancing threshold selection in tail modeling for risk management.

Findings

Explicit risk functions for new tests are derived.

Asymmetric weighting improves tail fit assessment.

Guidelines for choosing tests based on risk functions.

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.