Multivariate Discrete Outcomes: from Marginal Model Diagnostics to Nonparametric Copula Estimation

Multivariate discrete outcomes are common in a wide range of areas including insurance. When the interplay between outcomes is significant, quantifying dependencies among interrelated variables is of great importance. Due to their ability to flexibly accommodate dependence, copulas have been utilized extensively for dependence modeling in insurance. Yet the application of copulas on discrete data is still in its infancy. Although a substantial literature has emerged focusing on copula models under continuity, some key steps and concepts do not carry over to discrete data. One major barrier is the non-uniqueness of copulas, calling into question model interpretations and predictions. We study the issue of identifiability in a regression context and establish the conditions under which copula regression models are identifiable for discrete outcomes. Given identifiability, we propose a nonparametric estimator of copulas to identify the "hidden" dependence structure for discrete outcomes and develop its asymptotic properties. We explore the finite sample performance of our estimator under different scenarios using extensive simulation studies, and use our model to investigate the dependence of insurance claim frequencies across different business lines using a dataset from the Local Government Property Insurance Fund in the state of Wisconsin.

Beyond copula modeling, we extend some of the key concepts underlying our copula estimator for the purpose of regression diagnostics. Making informed decisions about model adequacy has long been an outstanding issue for discrete outcomes. To fill this gap, we develop an effective diagnostic tool for univariate regression models with discrete outcomes and show that it outperforms Pearson and deviance residuals for various diagnostic tasks.