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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 pp is much lager than sample size nn. The classical likelihood ratio test(LRT) is no longer applicable when pnp\gg n. Therefore a Quasi-LRT is proposed and asymptotic distribution of the test statistic under the null when p/n,np/n\rightarrow\infty, n\rightarrow\infty is well established in this paper. Meanwhile, John's test has been found to possess the powerful {\it dimension-proof} property, which keeps exactly the same limiting distribution under the null with any (n,p)(n,p)-asymptotic, i.e. p/n[0,]p/n\rightarrow[0,\infty], nn\rightarrow\infty. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison

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