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 limn→∞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.
Bibliographical noteFunding Information:
We would like to thank the reviewer for his/her constructive suggestions that have led to improvement ofthe paper. Chang’s research was supported in part by the Major Research Plan of the National Natural Science Foundation of China ( 91430108 ), the National Basic Research Program ( 2012CB955804 ), the National Natural Science Foundation of China ( 11171251 ), and the Major Program of Tianjin University of Finance and Economics ( ZD1302 ).
- Complete independence
- High dimension
- Limiting distribution
- Normal distribution