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Title: Testing for principal component directions under weak identifiability Authors:  Davy Paindaveine - Universite libre de Bruxelles (Belgium) [presenting]
Julien Remy - Universite libre de Bruxelles (Belgium)
Thomas Verdebout - Universite Libre de Bruxelles (Belgium)
Abstract: The problem of testing is considered which is on the basis of a p-variate Gaussian random sample, the null hypothesis $H_0: \theta_1= \theta_1^0$ against the alternative $H_1: \theta_1 \neq \theta_1^0$, where~$\theta_1$ is the "first" eigenvector of the underlying covariance matrix and $\theta_1^0$ is a fixed unit p-vector. In the classical setup where eigenvalues $\lambda_1>\lambda_2\geq \ldots\geq \lambda_p$ are fixed, the likelihood ratio test (LRT) and the Le Cam optimal test for this problem are asymptotically equivalent under the null, hence also under sequences of contiguous alternatives. We show that this equivalence does not survive asymptotic scenarios where $\lambda_{n1}-\lambda_{n2}=o(r_n)$ with $r_n=O(1/\sqrt{n})$. For such scenarios, the Le Cam optimal test still asymptotically meets the nominal level constraint, whereas the LRT becomes extremely liberal. Consequently, the former test should be favored over the latter one whenever the two largest sample eigenvalues are close to each other. By relying on the Le Cam theory of asymptotic experiments, we study in the aforementioned asymptotic scenarios the non-null and optimality properties of the Le Cam optimal test and show that the null robustness of this test is not obtained at the expense of efficiency. Our asymptotic investigation is extensive in the sense that it allows $r_n$ to converge to zero at an arbitrary rate.