## Bernoulli

• Bernoulli
• Volume 22, Number 2 (2016), 1184-1226.

### 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 $\widehat{\Sigma}$ 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.

#### Article information

Source
Bernoulli, Volume 22, Number 2 (2016), 1184-1226.

Dates
Revised: September 2014
First available in Project Euclid: 9 November 2015

https://projecteuclid.org/euclid.bj/1447077773

Digital Object Identifier
doi:10.3150/14-BEJ690

Mathematical Reviews number (MathSciNet)
MR3449812

Zentralblatt MATH identifier
06562309

#### Citation

Wegkamp, Marten; Zhao, Yue. Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas. Bernoulli 22 (2016), no. 2, 1184--1226. doi:10.3150/14-BEJ690. https://projecteuclid.org/euclid.bj/1447077773

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