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Title: Power enhancement for dimension detection of Gaussian signals Authors:  Gaspard Bernard - Universite Libre de Bruxelles (ULB) (Belgium) [presenting]
Thomas Verdebout - Universite Libre de Bruxelles (Belgium)
Abstract: The focus is on the classical problem of testing ${\cal H}_{0q}^{(n)}: \lambda_{q}^{(n)} > \lambda_{q+1}^{(n)}= \ldots= \lambda_{p}^{(n)}$, where $\lambda_{1}^{(n)}, \ldots, \lambda_{p}^{(n)}$ 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 ${\cal H}_{1q}^{(n)}: \lambda_{q}^{(n)} = \lambda_{q+1}^{(n)}= \ldots= \lambda_{p}^{(n)}$. This is very problematic if the latter procedure 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 to detect alternatives to the equality of the $p-q$ smallest roots of $\Sigma^{(n)}$ provided that $\lambda_{q}^{(n)}$ and $\lambda_{q+1}^{(n)}$ are such that $n^{1/2}(\lambda_{q}^{(n)} -\lambda_{q+1}^{(n)})$ diverges to $\infty$ as $n \rightarrow \infty$. We obtain tests ${\phi}_{\rm new}^{(n)}$ for the problem that keeps the local and asymptotic optimality properties of $\phi^{(n)}$ when $n^{1/2}(\lambda_{q}^{(n)} -\lambda_{q+1}^{(n)}) \to \infty$ and properly detect alternatives of the form ${\cal H}_{1q}^{(n)}$. Our results are illustrated via simulations and on a gene expression dataset from which we also discuss the problem of estimating the dimension of the signal.