Testing the sphericity of a covariance matrix when the dimension is much larger than the sample size

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sphericity test when the dimension p is much lager than sample size n. The classical likelihood ratio test(LRT) is no longer applicable when p 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 → ∞, n → ∞ 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 dimensionproof property, which keeps exactly the same limiting distribution under the null with any (n, p)-asymptotic, i.e. p/n → [0,∞], n → ∞. Nevertheless, the asymptotic distribution of both test statistics under the alternative hypothesis with a general population covariance matrix is also derived and incorporates the null distributions as special cases. The power functions are presented and proven to converge to 1 as p/n → ∞, n → ∞, n3/p = O(1). All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented to illustrate the finite sample performance of the results.