Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas

Abstract

We study the adaptive estimation of copula correlation matrix σ for the semi-parametric elliptical copula model. In this context, the correlations are connected to Kendall's tau through a sine function transformation. Hence, a natural estimate for σ is the plug-in estimator σσ with Kendall's tau statistic. We first obtain a sharp bound on the operator norm of σσ-σ. Then we study a factor model of σ, for which we propose a refined estimator σ by fitting a low-rank matrix plus a diagonal matrix to σσ using least squares with a nuclear norm penalty on the low-rank matrix. The bound on the operator norm of σσ-σ serves to scale the penalty term, and we obtain finite sample oracle inequalities for σ. We also consider an elementary factor copula model of σ, for which we propose closed-form estimators. All of our estimation procedures are entirely data-driven.