Coupled risk measures and their empirical estimation when losses follow heavy-tailed distributions

TL;DR

This paper develops a statistical inference framework for coupled risk measures, which are complex combinations of L-functionals, particularly applicable to heavy-tailed loss distributions, with implications for finance and economics.

Contribution

It introduces a novel inferential theory for coupled risk measures, extending existing results to ratios and complex combinations of L-functionals under heavy tails.

Findings

Provides asymptotic properties for estimators of coupled risk measures

Applies theory to distortion risk measures and economic inequality indices

Enables practical estimation for heavy-tailed loss data

Abstract

Considerable literature has been devoted to developing statistical inferential results for risk measures, especially for those that are of the form of L-functionals. However, practical and theoretical considerations have highlighted quite a number of risk measures that are of the form of ratios, or even more complex combinations, of two L-functionals. In the present paper we call such combinations `coupled risk measures' and develop a statistical inferential theory for them when losses follow heavy-tailed distributions. Our theory implies -at a stroke- statistical inferential results for absolute and relative distortion risk measures, weighted premium calculation principles, as well as for many indices of economic inequality that have appeared in the econometric literature.