Testing the Sphericity of a Covariance Matrix When the Dimension Is Much Larger Than the Sample Size

• Activity Number: 124
• Type: Contributed
• Date/Time: Monday, August 1, 2016 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Zeng Li*
• Companies: The University of Hong Kong
• Keywords: sphericity test ; ultra-large dimension ; John's test ; Quasi-likelihood ratio test

Abstract

This paper focuses on the prominent sphericity test when the dimension p is much lager than sample size n. The classical likelihood ratio test(LRT) is no longer applicable when p>>n. Therefore a Quasi-LRT is proposed and its asymptotic distribution of the test statistic under the null when p/n goes to infinity is well established in this paper. We also re-examine the well-known John's invariant test for sphericity in this ultra-dimensional setting. An amazing result from the paper states that John's test statistic has exactly the same limiting distribution under the ultra-dimensional setting with under other high-dimensional settings known in the literature. Therefore, John's test has been found to possess the powerful dimension-proof property, which keeps exactly the same limiting distribution under the null with any (n,p)-asymptotic, i.e. 0< p/n< infinity, n goes to infinity. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison.