Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas

Abstract

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 $\tilde{\Sigma }$ 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 $\tilde{\Sigma }$. 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.

Published at http://dx.doi.org/10.3150/14-BEJ690 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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