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 $p=p(n)\to \mathrm{\infty }$, $N=n+1\to \mathrm{\infty }$ and $p/n\to c\in (0,1)$, the asymptotic expansion and a uniform error bound of the distribution function of the logarithmic determinant of the sample correlation matrix $\log det(\hat{\mathit{R}})$ 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].

Keywords:

• Sample correlation matrix
• high-dimensional approximation
• Edgeworth expansion
• error bound

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (grant no. 11401169) and Foundation of He'nan educational committee (grant no. 20A110001).