Estimation of Copulas via Maximum Mean Discrepancy

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.

本文针对参数化Copula（copula）模型的鲁棒推断展开研究。采用典范最大似然估计进行参数推断时可能出现不稳定性，尤其在数据存在异常值的场景下。为此，本文提出一种基于最大平均差异（maximum mean discrepancy，MMD）准则的推断方法。我们推导了该新型估计量的非渐近预言不等式、相合性与渐近正态性。特别地，该预言不等式无需对Copula族施加任何假设，可应用于存在异常值或模型误设的情形。此外，在MMD框架下，针对那些相对于[0,1]^d上勒贝格测度不存在密度的Copula模型——如Marshall-Olkin Copula模型——的统计推断成为可能。仿真研究表明，本文提出的新方法具有良好的鲁棒性，相较伪最大似然估计的优势尤为显著。用于实现Copula模型MMD估计量的R包已公开上线。本文的补充材料可在线获取。