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].

本文考虑了具有独立分量的高斯种群高维样本相关矩阵的行列式的渐近分布。特别地，当尺寸p和样本大小N满足时$p=p（ñ）\to \mathrm{\infty }$， $ñ=ñ+\mathrm{1个}\to \mathrm{\infty }$ 和 $p/ñ\to C\in （0，\mathrm{1个}）$，样本相关矩阵对数行列式的分布函数的渐近展开和一致误差界 $\mathrm{日志}dËŤ（\widehat{\mathit{[R}}）$通过Edgeworth扩展方法获得。结果以高维独立试验中的应用还提出，一些数值模拟表明，该方法优于传统的卡方近似法和进行高效如姜和羊[介绍的方法中心极限定理古典似然高维正态分布的比率测试，Ann。统计员。41（4）（2013），第2029–2074页]。