Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas

We study the adaptive estimation of copula correlation matrix Sigma 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 Sigma is the plug-in estimator (Sigma) over cap with Kendall's tau statistic. We first obtain a sharp bound on the operator norm of (Sigma) over cap - Sigma. Then we study a factor model of Sigma, for which we propose a refined estimator (Sigma) over tilde by fitting a low-rank matrix plus a diagonal matrix to (Sigma) over cap using least squares with a nuclear norm penalty on the low-rank matrix. The bound on the operator norm of (Sigma) over cap - Sigma serves to scale the penalty term, and we obtain finite sample oracle inequalities for (Sigma) over tilde. We also consider an elementary factor copula model of Sigma, for which we propose closed-form estimators. All of our estimation procedures are entirely data-driven.