Testing for subsphericity when $n$ and $p$ are of different asymptotic order

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 $p/n \rightarrow 0$ or $p/n \rightarrow \infty$. 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 $p/n \rightarrow \gamma \in (0, \infty)$. Simulations are used to demonstrate the results, providing also evidence that the test might be further extendable to a wider asymptotic setting.