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 \hat{\Sigma} with Kendall's tau statistic. We first obtain a sharp bound on the operator norm of \hat{\Sigma} \Sigma$ Then we study a factor model of \Sigma$ for which we propose a refined estimator \widetilde{\Sigma} by fitting a low rank matrix plus a diagonal matrix to \hat{\Sigma} using least squares with a nuclear norm penalty on the low rank matrix. The bound on the operator norm of \hat{\Sigma} \Sigma$ serves to scale the penalty term, and we obtain finite sample oracle inequalities for \widetilde{\Sigma} 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.