Moderate Deviation Principle for the Determinant of Sample Correlation Matrix.

Let X1,…,Xn$$ {\mathbf{X}}_1,\dots, {\mathbf{X}}_n $$ be a sequence of independence random vectors from Np(μ,Σ)$$ {N}_p\left(\boldsymbol{\mu}, \boldsymbol{\varSigma} \right) $$ with population correlation matrix Rn$$ {\mathbf{R}}_n $$, and its sample correlation matrix is denoted as R^n=(r^ij)p×p$$ {\hat{\mathbf{R}}}_n={\left({\hat{r}}_{ij}\right)}_{p\times p} $$, where the matrix element r^ij$$ {\hat{r}}_{ij} $$ represents the Pearson correlation coefficient between the i$$ i $$ and j$$ j $$ columns of the data matrix (X1,…,Xn)′$$ {\left({\mathbf{X}}_1,\dots, {\mathbf{X}}_n\right)}^{\prime } $$. R^n$$ {\hat{\mathbf{R}}}_n $$ is an interesting part in multivariate analysis, and the likelihood ratio test (LRT) statistic used to test the complete independence of all components of X$$ \mathbf{X} $$ can be expressed as a function of R^n$$ {\hat{\mathbf{R}}}_n $$. On this basis, under the condition that the sample size n$$ n $$ and the dimension p$$ p $$ satisfy p/n→y∈(0,1]$$ p/n\to y\in \left(0,1\right] $$ as n→∞$$ n\to \infty $$, we derive the moderate deviation principle (MDP) of the LRT statistic for the hypothesis testing problem H0:Rn=Ip$$ {H}_0:{\mathbf{R}}_n={\mathbf{I}}_p $$ vs H1:Rn≠Ip$$ {H}_1:{\mathbf{R}}_n\ne {\mathbf{I}}_p $$. [ABSTRACT FROM AUTHOR]