High-dimensional Edgeworth expansion of the determinant of sample correlation matrix and its error bound

Abstract

The paper considers the asymptotic distribution on the determinant of the high-dimensional sample correlation matrix of the Gaussian population with independent components. In particular, when the dimension p and the sample size N satisfy , and , the asymptotic expansion and a uniform error bound of the distribution function of the logarithmic determinant of the sample correlation matrix are obtained by the Edgeworth expansion method. An application of the result to high-dimensional independence test is also proposed, some numerical simulations reveal that the proposed method outperforms the traditional chi-square approximation method and performs as efficient as the method introduced by Jiang and Yang [Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions, Ann. Statist. 41(4) (2013), pp. 2029–2074].