Assessing Risk Heterogeneity through Heavy-Tailed Frequency and Severity Mixtures

TL;DR

This paper develops a canonical framework to quantify risk heterogeneity by modeling frequency and severity as heavy-tailed mixtures built from Negative Binomial and Gamma kernels. It leverages a NB–Gamma duality to derive necessary and sufficient conditions for heavy-tailed mixtures and constructs flexible 4- and 5-parameter families (via Geometric and Exponential kernels, extended to arbitrary kernels) to detect and measure heterogeneity. A key contribution is the introduction of calibrated mixing families, which allow robustness checks by re-expressing fitted models with alternative kernels while preserving the same empirical fit. The framework enables explicit identification of high-risk subpopulations and tail risk in insurance and related fields, with practical steps for empirical application and robustness assessment through the HGZY, ZY, and generalized SigmaBeta families and their calibrated counterparts.

Abstract

The analysis of risk typically involves dividing a random damage-generation process into separate frequency (event-count) and severity (damage-magnitude) components. In the present article, we construct canonical families of mixture distributions for each of these components, based on a Negative Binomial kernel for frequencies and a Gamma kernel for severities. These mixtures are employed to assess the heterogeneity of risk factors underlying an empirical distribution through the shape of the implied mixing distribution. From the duality of the Negative Binomial and Gamma distributions, we first derive necessary and sufficient conditions for heavy-tailed (i.e., inverse power-law) canonical mixtures. We then formulate flexible 4-parameter families of mixing distributions for Geometric and Exponential kernels to generate heavy-tailed 4-parameter mixture models, and extend these mixtures to arbitrary Negative Binomial and Gamma kernels, respectively, yielding 5-parameter mixtures for detecting and measuring risk heterogeneity. To check the robustness of such heterogeneity inferences, we show how a fitted 5-parameter model may be re-expressed in terms of alternative Negative Binomial or Gamma kernels whose associated mixing distributions form a "calibrated" family.