Analysis of the copula correlation matrix for meta-elliptical distributions

We study the 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 statistics. In this talk, we first obtain a sharp bound on 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 on the operator norm $\widehat \Sigma - \Sigma$ serves to scale the penalty term, and we obtain finite sample oracle inequalities for $\widetilde\Sigma$. If time permits, we may also present two estimators based on suitably truncated eigen-decompositions of $\widehat\Sigma$, one for an elementary factor model and the other for the regime where $d$ is proportional to the sample size. (with Marten Wegkamp)