Statistics and Its Interface

Volume 7 (2014)

Number 2

Optimal estimation of sparse correlation matrices of semiparametric Gaussian copulas

Pages: 201 – 209

DOI: http://dx.doi.org/10.4310/SII.2014.v7.n2.a5

Authors

Lingzhou Xue (Department of Operations Research, and Financial Engineering, Princeton University, Princeton, New Jersey, U.S.A.)

Hui Zou (School of Statistics, University of Minnesota, Minneapolis, Mn., U.S.A.)

Abstract

Statistical inference of semiparametric Gaussian copulas is well studied in the classical fixed dimension and large sample size setting. Nevertheless, optimal estimation of the correlation matrix of semiparametric Gaussian copula is understudied, especially when the dimension can far exceed the sample size. In this paper we derive the minimax rate of convergence under the matrix $\ell_1$-norm and $\ell_2$-norm for estimating large correlation matrices of semiparametric Gaussian copulas when the correlation matrices are in a weak $\ell_q$ ball. We further show that an explicit rank-based thresholding estimator adaptively attains minimax optimal rate of convergence simultaneously for all $0 \leq q \lt 1$. Numerical examples are provided to demonstrate the finite sample performance of the rank-based thresholding estimator.

Keywords

correlation matrix, Gaussian copula, minimax optimality, rank correlation, thresholding, weak $\ell_q$ ball

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