Analysis of elliptical copula correlation factor model with Kendall's tau

We study a factor model for the correlation matrix $\Sigma\in\RR^{d\times d}$ of an elliptical copula. The correlations are connected to Kendall's tau and a natural estimation procedure is to plug-in Kendall's tau statistics. The resulting matrix $\wh \Sigma$ can be viewed as a preliminary estimator of $\Sigma$ and we obtain sharp bound on the operator norm of $\wh \Sigma - \Sigma$. We propose a refined estimator $\widetilde{\Sigma}$ of $\Sigma$ by fitting a low-rank matrix plus a diagonal matrix to $\wh \Sigma$ using least squares with a nuclear norm penalty on the low-rank matrix. We obtain finite sample oracle inequalities for $\widetilde{\Sigma}$. In addition, we present two estimators based on suitably truncated eigen-decompositions of $\wh\Sigma$, one for an elementary factor model and the other for the regime where $d$ is proportional to the sample size.