Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas
This work addresses statistical estimation challenges in copula modeling for applications like finance and risk analysis, representing an incremental improvement with specific methodological refinements.
The paper tackles the problem of estimating the copula correlation matrix in semiparametric elliptical copulas, developing plug-in and refined estimators based on Kendall's tau and factor models, with results including sharp bounds on operator norms and finite sample oracle inequalities.
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.