Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas

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