On Tests for Complete Independence of Normal Random Vectors

TL;DR

This paper analyzes the asymptotic behavior of two tests for complete independence in high-dimensional normal vectors, showing their convergence to normal and chi-square distributions under certain conditions.

Contribution

It provides a detailed investigation of the limiting distributions of Schott's and Mao's test statistics, extending understanding of their asymptotic properties in high-dimensional settings.

Findings

Both test statistics converge to the standard normal distribution after normalization.

The distribution functions can be well approximated by chi-square distributions with p(p-1)/2 degrees of freedom.

Results hold as both n and p tend to infinity, with p/n approaching a finite limit.

Abstract

Consider a random sample of $n$ independently and identically distributed $p$-dimensional normal random vectors. A test statistic for complete independence of high-dimensional normal distributions, proposed by Schott (2005), is defined as the sum of squared Pearson's correlation coefficients. A modified test statistic has been proposed by Mao (2014). Under the assumption of complete independence, both test statistics are asymptotically normal if the limit $\lim_{n\to\infty}p/n$ exists and is finite. In this paper, we investigate the limiting distributions for both Schott's and Mao's test statistics. We show that both test statistics, after suitably normalized, converge in distribution to the standard normal as long as both $n$ and $p$ tend to infinity. Furthermore, we show that the distribution functions of the test statistics can be approximated very well by a chi-square distribution function with $p(p-1)/2$ degrees of freedom as $n$ tends to infinity regardless of how $p$ changes with $n$.