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Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas

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