  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 case $p/n \rightarrow
\gamma \in (0, \infty)$. As our second main contribution, we use the test to
derive a consistent estimator for the latent dimension of the model.
Simulations and a real data example are used to demonstrate the results,
providing also evidence that the test might be further extendable to a wider
asymptotic regime.
