Nonparametric Copula Estimation for Mixed Insurance Claim Data

Abstract

Multivariate claim data are common in insurance applications, for example, claims of each policyholder from different types of insurance coverages. Understanding the dependencies among such multivariate risks is critical to the solvency and profitability of insurers. Effectively modeling insurance claim data is challenging due to their special complexities. At the policyholder level, claim outcomes usually follow a two-part mixed distribution: a probability mass at zero corresponding to no claim and an otherwise positive claim from a skewed and long-tailed distribution. To simultaneously accommodate the complex features of the marginal distributions while flexibly quantifying the dependencies among multivariate claims, copula models are commonly used. Although a substantial body of literature focusing on copulas with continuous outcomes has emerged, some key steps do not carry over to mixed data. In particular, existing nonparametric copula estimators are not consistent for mixed data, and thus copula specification and diagnostics for mixed outcomes have been a problem. However, insurance is a closely regulated industry in which model validation is particularly important, and it is essential to develop a baseline nonparametric copula estimator to identify the underlying dependence structure. In this article, we fill in this gap by developing a nonparametric copula estimator for mixed data. We show the uniform convergence of the proposed nonparametric copula estimator. Through simulation studies, we demonstrate that the proportion of zeros plays a key role in the finite sample performance of the proposed estimator. Using the claim data from the Wisconsin Local Government Property Insurance Fund, we illustrate that our nonparametric copula estimator can assist analysts in identifying important features of the underlying dependence structure, revealing how different claims or risks are related to one another.