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RANK TESTS FOR PCA UNDER WEAK IDENTIFIABILITY
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DOI：10.1214/25-AOS2552.png)
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Abstract
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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) against the alternative hypothesis H-1 : theta not equal 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 carefully 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. This requires original results on quadratic mean differentiable families for triangular arrays of observations, which are likely to be of interest in other models, too. Even in non-LAN experiments, our results enable us to investigate, through Le Cam's first and third lemmas, the asymptotic null and nonnull 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 outstanding uniform efficiency properties under such nonstandard scenarios. In particular, Gaussian-score rank tests, under arbitrarily weak identifiability, still uniformly dominate their parametric pseudo-Gaussian competitor in terms of asymptotic relative efficiencies. Our theoretical results, which are the first ones to study rank tests in the triangular array framework allowing for weak identifiability, are supported by several Monte Carlo exercises.
Keywords:
Elliptical densities
limiting experiments
multivariate signs and ranks
principal component analysis
spiked scatter matrices
triangular arrays of observations
weak identifiability
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