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

TL;DR

This paper extends a classical subsphericity test to high-dimensional settings where the number of variables and sample size grow at different rates, and uses it to estimate the latent dimension of the covariance matrix.

Contribution

It generalizes the subsphericity test to regimes with diverging eigenvalues and different asymptotic ratios of p to n, and introduces a consistent estimator for the latent dimension.

Findings

The test performs well in simulations under various asymptotic regimes.

The estimator accurately recovers the latent dimension in real data examples.

Results suggest potential for extending the test to broader asymptotic settings.

Abstract

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.