High-dimensional Edgeworth expansion of the determinant of sample correlation matrix and its error bound. Issue 3 (3rd April 2021)

• Record Type: Journal Article
• Authors: Xie, Junshan; Sun, Gaoming
• Is Part Of: Stochastics. Volume 93:Issue 3(2021)
• Page Start: 428
• Page End: 446
• Publication Date: 2021-04-03
• Subjects: Sample correlation matrix -- high-dimensional approximation -- Edgeworth expansion -- error bound; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2020.1744604
• Languages: English
• ISSNs: 1744-2508

Abstract

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 ) → ∞, N = n + 1 → ∞ and p / n → c ∈ ( 0, 1 ), the asymptotic expansion and a uniform error bound of the distribution function of the logarithmic determinant of the sample correlation matrix log ⁡ d e t ( 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].