Multivariate discrete outcomes are common in a wide range of areas including insurance, finance, and biology. When the interplay between outcomes is significant, quantifying dependencies among interrelated variables is of great importance. Due to their ability to accommodate dependence flexibly, copulas are being applied increasingly. Yet, the application of copulas on discrete data is still in its infancy; one of the biggest barriers is the nonuniqueness of copulas, calling into question model interpretations and predictions. In this article, we study copula estimation with discrete outcomes in a regression context. As the marginal distributions vary with covariates, inclusion of continuous regressors expands the region of support for consistent estimation of copulas. Because some properties of continuous outcomes do not carry over to discrete outcomes, specification of a copula model has been a problem. We propose a nonparametric estimator of copulas to identify the “hidden” dependence structure for discrete outcomes and develop its asymptotic properties. The proposed nonparametric estimator can also serve as a diagnostic tool for selecting a parametric form for copulas. In the simulation study, we explore the performance of the proposed estimator under different scenarios and provide guidance on when the choice of copulas is important. The performance of the estimator improves as discreteness diminishes. A practical bandwidth selector is also proposed. An empirical analysis examines a dataset from the Local Government Property Insurance Fund (LGPIF) in the state of Wisconsin. We apply the nonparametric estimator to model the dependence among claim frequencies from different types of insurance coverage. Supplementary materials for this article are available online.
Bibliographical noteFunding Information:
The authors are grateful to the reviewers for insightful comments leading to an improved article. The work of the first two authors (Yang and Frees) was partially funded by a Society of Actuaries? Center of Actuarial Excellence Grant.
- Copula specification
- Insurance claim frequency