Rank tests for PCA under weak identifiability

Davy Paindaveine

Université libre de Bruxelles

Friday 25th October 2024

12:00:00

Aula Saleri - Seminari VI piano

This talk considers a testing problem that is associated with principal component analysis and that may be of interest whenever dimension reduction is considered, as it is the case, e.g., when analyzing high-dimensional data in life sciences or environmental sciences. More specifically, in a triangular array framework where n observations are randomly sampled from a p-dimensional elliptical distribution with shape matrix V_n, we consider the problem of testing the null hypothesis H_0: theta=theta_0, where theta is the (fixed) leading unit eigenvector of V_n and theta_0 is a given unit p-vector. The dependence of the shape matrix on the sample size allows us to consider challenging asymptotic scenarios in which the parameter of interest theta is unidentified in the limit, because the ratio between both leading eigenvalues of V_n converges to one. We study the corresponding limiting experiments under such weak identifiability, and we show that these may be LAN or non-LAN. While earlier work in the framework was strictly limited to Gaussian distributions, where the study of local log-likelihood ratios could simply rely on explicit expressions, our asymptotic investigation allows for essentially arbitrary elliptical distributions. Even in non-LAN experiments, our results enable us to investigate the asymptotic null and non-null properties of multivariate rank tests. These nonparametric tests are shown to exhibit an excellent behavior under weak identifiability: not only do they maintain the target nominal size irrespective of the amount of weak identifiability, but they also keep their uniform efficiency properties under such non-standard scenarios. This initiative is part of the “Ph.D. Lectures” activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to Ph.D. students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk. Contatti: laura.sangalli@polimi.it