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.

翻译：我们根据样本共变矩阵的二元值的前两个时刻,对亚球性进行古典试验。我们把这种试验推广到高维系统,在高维系统中,共变矩阵的信号值与无穷无尽相区别,或者美元/n\rightrow 0$,或者美元/p/n\rightrowr\infty$。在后一种情况下,我们进一步要求从某种具体意义上说,电子值的差异是适当的。我们的工作可以被看成是对Schott(2006年)的工作的补充,Schott(2006年)在案件(0.00,\infty)中建立了等值的结果。作为我们的第二个主要贡献,我们使用测试来得出模型潜在维度的一致的估算值。我们使用了模拟和真实数据示例来展示结果,同时提供证据,证明测试可能进一步扩展到范围更广的微调制度。