Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas

We study the adaptive estimation of copula correlation matrix $\Sigma$ for elliptical copulas. 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 $\widehat\Sigma$ with Kendall's tau statistic. We first obtain a sharp bound for the operator norm of $\widehat \Sigma - \Sigma$. Then, we study a factor model for $\Sigma$, for which we propose a refined estimator $\widetilde\Sigma$ by fitting a low-rank matrix plus a diagonal matrix to $\widehat\Sigma$ using least squares with a nuclear norm penalty on the low-rank matrix. The bound for the operator norm of $\widehat \Sigma - \Sigma$ serves to scale the penalty term, and we obtain finite sample oracle inequalities for $\widetilde\Sigma$. We also consider an elementary factor model of $\Sigma$, for which we propose closed-form estimators. We provide data-driven versions for all our estimation procedures and performance bounds.