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February 2020 Testing for principal component directions under weak identifiability
Davy Paindaveine, Julien Remy, Thomas Verdebout
Ann. Statist. 48(1): 324-345 (February 2020). DOI: 10.1214/18-AOS1805

Abstract

We consider the problem of testing, on the basis of a $p$-variate Gaussian random sample, the null hypothesis $\mathcal{H}_{0}:\boldsymbol{\theta}_{1}=\boldsymbol{\theta}_{1}^{0}$ against the alternative $\mathcal{H}_{1}:\boldsymbol{\theta}_{1}\neq \boldsymbol{\theta}_{1}^{0}$, where $\boldsymbol{\theta}_{1}$ is the “first” eigenvector of the underlying covariance matrix and $\boldsymbol{\theta}_{1}^{0}$ is a fixed unit $p$-vector. In the classical setup where eigenvalues $\lambda_{1}>\lambda_{2}\geq \cdots \geq \lambda_{p}$ are fixed, the Anderson (Ann. Math. Stat. 34 (1963) 122–148) likelihood ratio test (LRT) and the Hallin, Paindaveine and Verdebout (Ann. Statist. 38 (2010) 3245–3299) 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 $\lambda_{n1}/\lambda_{n2}=1+O(r_{n})$ 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 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 asymptotic investigation is extensive in the sense that it allows $r_{n}$ to converge to zero at an arbitrary rate. While we restrict to single-spiked spectra of the form $\lambda_{n1}>\lambda_{n2}=\cdots =\lambda_{np}$ to make our results as striking as possible, we extend our results to the more general elliptical case. Finally, we present an illustrative real data example.

Citation

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Davy Paindaveine. Julien Remy. Thomas Verdebout. "Testing for principal component directions under weak identifiability." Ann. Statist. 48 (1) 324 - 345, February 2020. https://doi.org/10.1214/18-AOS1805

Information

Received: 1 October 2017; Revised: 1 July 2018; Published: February 2020
First available in Project Euclid: 17 February 2020

zbMATH: 07196541
MathSciNet: MR4065164
Digital Object Identifier: 10.1214/18-AOS1805

Subjects:
Primary: 62F05, 62H25
Secondary: 62E20

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 1 • February 2020
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