Testing for Principal Component Directions under Weak Identifiability

We consider the problem of testing, 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 \thetab_1^0, where \thetab_1 is the "first" eigenvector of the underlying covariance matrix and \thetab_1^0 is a fixed unit p-vector. In the classical setup where eigenvalues \lambda_1>\lambda_2\geq .\geq \lambda_p are fixed, the Anderson (1963) likelihood ratio test (LRT) and the Hallin, Paindavine and Verdebout (2010) 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. To make our results as striking as possible, we not only restrict to the multinormal case but also to single-spiked spectra of the form \lambda_{n1}>\lambda_{n2}=.=\lambda_{np}.