Large deviations for risk measures in finite mixture models

TL;DR

This paper investigates the asymptotic error behavior of estimated risk measures in finite mixture models, providing large deviation results and explicit formulas for common risk measures like quantiles and Expected Shortfall.

Contribution

It introduces a large deviations framework for analyzing the error in estimated risk measures in finite mixture models, with explicit formulas for key risk measures.

Findings

Large deviation principles are established for estimated risk measures.

Explicit rate functions are derived for mixtures of two models.

Results are applied to quantiles, Expected Shortfall, and shortfall risk measures.

Abstract

Due to their heterogeneity, insurance risks can be properly described as a mixture of different fixed models, where the weights assigned to each model may be estimated empirically from a sample of available data. If a risk measure is evaluated on the estimated mixture instead of the (unknown) true one, then it is important to investigate the committed error. In this paper we study the asymptotic behaviour of estimated risk measures, as the data sample size tends to infinity, in the fashion of large deviations. We obtain large deviation results by applying the contraction principle, and the rate functions are given by a suitable variational formula; explicit expressions are available for mixtures of two models. Finally, our results are applied to the most common risk measures, namely the quantiles, the Expected Shortfall and the shortfall risk measures.