Minimum Copula Divergence for Robust Estimation

Abstract: This paper introduces a robust estimation framework based solely on the copula function. We begin by introducing a family of divergence measures tailored for copulas, including the (\alpha)-, (\beta)-, and (\gamma)-copula divergences, which quantify the discrepancy between a parametric copula model and an empirical copula derived from data independently of marginal specifications. Using these divergence measures, we propose the minimum copula divergence estimator (MCDE), an estimation method that minimizes the divergence between the model and the empirical copula. The framework proves particularly effective in addressing model misspecifications and analyzing heavy-tailed data, where traditional methods such as the maximum likelihood estimator (MLE) may fail. Theoretical results show that common copula families, including Archimedean and elliptical copulas, satisfy conditions ensuring the boundedness of divergence-based estimators, thereby guaranteeing the robustness of MCDE, especially in the presence of extreme observations. Numerical examples further underscore MCDE's ability to adapt to varying dependence structures, ensuring its utility in real-world scenarios.