Testing for subsphericity when n and p are of different asymptotic order
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In this short note, we extend a classical test of subsphericity, based on the first two moments of the eigenvalues of the sample covariance matrix, to the high-dimensional regime where the signal eigenvalues of the covariance matrix diverge to infinity and either p/n → 0 or p/n →∞. In the latter case we further require that the divergence of the eigenvalues is suitably fast in a specific sense. Our work can be seen to complement that of Schott (2006) who established equivalent results in the regime p/n →γ∈ (0, ∞). Simulations are used to demonstrate the results, providing also evidence that the test might be further extendable to a wider asymptotic setting.
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