Testing for Principal Component Directions Under Weak Identifiability

• Activity Number: 281 - The State-of-the-Art Developments with Nonignorable Nonresponse
• Type: Invited
• Date/Time: Wednesday, August 5, 2020 : 10:00 AM to 11:50 AM
• Sponsor: Survey Research Methods Section
• Author(s): Davy Paindaveine* and Thomas Verdebout and Julien Remy
• Companies: Université libre de Bruxelles and Université Toulouse Capitole and Université libre de Bruxelles and Université libre de Bruxelles
• Keywords: Local asymptotic normality; Principal component analysis; Spiked covariance matrices; Weak identifiability

Abstract

We consider the problem of testing the null hypothesis that the first principal direction coincides with a given direction in the multivariate Gaussian model. In the classical setup where eigenvalues are fixed, the likelihood ratio test (LRT) and the Le Cam optimal test for this problem are asymptotically equivalent under the null hypothesis, hence also under sequences of contiguous alternatives. We show that this equivalence does not survive asymptotic scenarios where the ratio of both leading eigenvalues goes to one faster than the inverse of root-n. For such scenarios, the Le Cam optimal test still asymptotically meets the nominal level constraint, whereas the LRT severely overrejects the null hypothesis. 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's asymptotic theory of statistical experiments, we study the non-null and optimality properties of the Le Cam optimal test in the aforementioned asymptotic scenarios and show that the null robustness of this test is not obtained at the expense of power. Our results are illustrated on a real data example.