Power enhancement for dimension detection of Gaussian signals

Abstract

We consider the classical problem of testing ${\mathcal{H}}_{0q}^{(n)};{\lambda }_q^{(n)}>{\lambda }_{q+1}^{(n)}=\cdots ={\lambda }_p^{(n)}$, where ${\lambda }_1^{(n)},\cdots ,{\lambda }_p^{(n)}$ are the ordered latent roots of covariance matrices ${\mathrm{\Sigma }}^{(n)}$. We show that the usual Gaussian procedure, ${\varphi }^{(n)}$, for this problem essentially shows no power against alternatives of weaker signals of the form ${\mathcal{H}}_{1q}^{(n)};{\lambda }_q^{(n)}={\lambda }_{q+1}^{(n)}=\cdots ={\lambda }_p^{(n)}$, which is problematic if it is used to perform inference on the true dimension of the signal. We show that the same test ${\varphi }^{(n)}$ enjoys some local and asymptotic optimality properties for detecting alternatives to the equality of the $p-q$ smallest roots of ${\mathrm{\Sigma }}^{(n)}$, provided that ${\lambda }_q^{(n)}$ and ${\varphi }_{q+1}^{(n)}$ are sufficiently separated. We obtain tests, ${\varphi }_{new}^{(n)}$, for the problem that retain the local and asymptotic optimality properties of ${\varphi }^{(n)}$ when ${\lambda }_q^{(n)}$ and ${\lambda }_{q+1}^{(n)}$ are sufficiently separated and properly detect alternatives of the form ${\mathcal{H}}_{1q}^{(n)}$ . 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.