Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas
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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 Σ̂ with Kendall's tau statistic. We first obtain a sharp bound on the operator norm of Σ̂-Σ. Then we study a factor model of Σ, for which we propose a refined estimator Σ by fitting a low-rank matrix plus a diagonal matrix to Σ̂ using least squares with a nuclear norm penalty on the low-rank matrix. The bound on the operator norm of Σ̂-Σ serves to scale the penalty term, and we obtain finite sample oracle inequalities for Σ. 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.
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