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May 2016 Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas
Marten Wegkamp, Yue Zhao
Bernoulli 22(2): 1184-1226 (May 2016). DOI: 10.3150/14-BEJ690

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 $\widehat{\Sigma}$ with Kendall’s tau statistic. We first obtain a sharp bound on the operator norm of $\widehat{\Sigma}-\Sigma$. Then we study a factor model of $\Sigma$, for which we propose a refined estimator $\widetilde{\Sigma}$ by fitting a low-rank matrix plus a diagonal matrix to $\widehat{\Sigma}$ using least squares with a nuclear norm penalty on the low-rank matrix. The bound on the operator norm of $\widehat{\Sigma}-\Sigma$ serves to scale the penalty term, and we obtain finite sample oracle inequalities for $\widetilde{\Sigma}$. 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.

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Marten Wegkamp. Yue Zhao. "Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas." Bernoulli 22 (2) 1184 - 1226, May 2016. https://doi.org/10.3150/14-BEJ690

Information

Received: 1 January 2014; Revised: 1 September 2014; Published: May 2016
First available in Project Euclid: 9 November 2015

zbMATH: 06562309
MathSciNet: MR3449812
Digital Object Identifier: 10.3150/14-BEJ690

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

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Vol.22 • No. 2 • May 2016
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