Estimation of copulas via Maximum Mean Discrepancy

TL;DR

This paper introduces a robust copula estimation method based on Maximum Mean Discrepancy (MMD), providing theoretical guarantees and improved robustness over traditional likelihood-based approaches, especially with outliers or non-dense copulas.

Contribution

It develops a new MMD-based estimator for copulas, with proven consistency, asymptotic normality, and applicability to non-dense copulas, enhancing robustness and theoretical understanding.

Findings

The MMD estimator is robust to outliers.

It outperforms pseudo-maximum likelihood in simulations.

The method applies to non-dense copulas like Marshall-Olkin.

Abstract

This paper deals with robust inference for parametric copula models. Estimation using Canonical Maximum Likelihood might be unstable, especially in the presence of outliers. We propose to use a procedure based on the Maximum Mean Discrepancy (MMD) principle. We derive non-asymptotic oracle inequalities, consistency and asymptotic normality of this new estimator. In particular, the oracle inequality holds without any assumption on the copula family, and can be applied in the presence of outliers or under misspecification. Moreover, in our MMD framework, the statistical inference of copula models for which there exists no density with respect to the Lebesgue measure on $[0,1]^d$, as the Marshall-Olkin copula, becomes feasible. A simulation study shows the robustness of our new procedures, especially compared to pseudo-maximum likelihood estimation. An R package implementing the MMD estimator for copula models is available.