Abstract
We consider the classical problem of testing $\mathcal{H}^{(n)}{0q} : \lambda^{(n)}{q}>\lambda^{(n)}{q+1}=\cdots=\lambda^{(n)}{p}$, where $\lambda^{(n)}{1},\cdots,\lambda^{(n)}p$ are the ordered latent roots of covariance matrices $\Sigma^{(n)}$. We show that the usual Gaussian procedure, $\phi^{(n)}$, for this problem essentially shows no power against alternatives of weaker signals of the form $\mathcal{H}^{(n)}{1q}$ : $\lambda^{(n)}{q} = \lambda^{(n)}{q+1}=\cdots=\lambda^{(n)}{p}$, which is problematic if it is used to perform inference on the true dimension of the signal. We show that the same test $\phi^{(n)}$ enjoys some local and asymptotic optimality properties for detecting alternatives to the equality of the $p-q$ smallest roots of $\Sigma^{(n)}$, provided that $\lambda^{(n)}{q}$ and $\phi^{(n)}{q+1}$ are sufficiently separated. We obtain tests, $\phi^{(n)}{new}$, for the problem that retain the local and asymptotic optimality properties of $\phi^{(n)}$ when $\lambda^{(n)}{q}$ and $\lambda^{(n)}{q+1}$ are sufficiently separated and properly detect alternatives of the form $\mathcal{H}^{(n)}{1q}$ . We illustrate the performances of our tests using simulations and on a gene expression data set, where we also discuss the problem of estimating the dimension of the signal.
