A test for the complete independence of high-dimensional random vectors

ABSTRACT

This paper discusses the problem of testing the complete independence of random variables when the dimension of observations can be much larger than the sample size. It is reported that two typical tests based on, respectively, the biggest off-diagonal entry and the largest eigenvalue of the sample correlation matrix lose their control of type I error in such high-dimensional scenarios, and exhibit distinct behaviours in type II error under different types of alternative hypothesis. Given these facts, we propose a permutation test procedure by synthesizing these two extreme statistics. Simulation results show that for finite dimension and sample size the proposed test outperforms the existing methods in various cases.

KEYWORDS:

• Correlation matrix
• extreme value distribution
• high-dimensional
• hypothesis testing
• permutation test

AMS SUBJECT CLASSIFICATION:

• 05A05
• 62H15
• 15A18

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Weiming Li's research is supported by National Natural Science Foundation of China, No. 11401037. Zhi Liu's research is supported by FDCT078/2013/A3 of Macau.