Estimation of Copulas via Maximum Mean Discrepancy

Abstract

This article 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 nonasymptotic 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. Supplementary materials for this article are available online.

KEYWORDS:

• Algorithms semiparametric inference
• Copula
• Kernel methods and RKHS
• Robust procedures

Supplementary Materials

Plot 3D Marshall-Olkin.html: contains the interactive plot of the MSE for the Marshall-Olkin family of copulas. Plot 3D parametric families.html: contains the interactive plot of the MSE for parametric families of copulas. Reproducibility: this folder contains the code and instructions to reproduce the figures of the article.

Acknowledgments

The authors thank both anonymous Referees for their insightful comments that led to many improvements of the article.

Funding

Additional information

Funding

Badr-Eddine Chérief-Abdellatif acknowledges support of the UK Defence Science and Technology Laboratory (DSTL) and EPSRC under grant EP/R013616/1. This is part of the collaboration between US DOD, UK MOD and UK EPSRC under the Multidisciplinary University Research Initiative. Jean-David Fermanian has been supported by the labex Ecodec (reference project ANR-11-LABEX-0047).