High-dimensional asymptotic expansion of the null distribution for Schott's test statistic for complete independence of normal random variables

• Published in: Communications in statistics. Theory and methods Vol. 53; no. 3; pp. 909 - 925
• Main Author: Yamada, Takayuki
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 01-02-2024; Taylor & Francis Ltd
• Subjects: Asymptotic expansion of the null distribution; Asymptotic series; complete independence; Critical point; high-dimensional data; Infinity; p)-asymptotic; Primary 62H30; Random variables; Secondary 62H12
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1080/03610926.2022.2094414

This article is concerned with the testing complete independence for the elements of observed vector. Schott proposed the testing statistic T and gave limiting null distribution under the high-dimensional asymptotic framework that the sample size n and the dimensionality p go to infinity together while p/n converges to a positive constant. In this article we give a one-term asymptotic expansion of the null distribution for T as
min
{
n
,
p
}
tends toward infinity. We derive a correction of the critical point for Schott's test based on this expansion. The finite sample size and dimensionality performance for attained significance level is evaluated in a simulation study and the results are compared to those of Schott's test.