Optimal estimation of sparse correlation matrices of semiparametric Gaussian copulas

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 l₁-norm and l₂-norm for estimating large correlation matrices of semiparametric Gaussian copulas when the correlation matrices are in a weak l_q ball. We further show that an explicit rank-based thresholding estimator adaptively attains minimax optimal rate of convergence simultaneously for all 0 ≤ q < 1. Numerical examples are provided to demonstrate the finite sample performance of the rank-based thresholding estimator.