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November 2020 Sign tests for weak principal directions
Davy Paindaveine, Julien Remy, Thomas Verdebout
Bernoulli 26(4): 2987-3016 (November 2020). DOI: 10.3150/20-BEJ1213

Abstract

We consider inference on the first principal direction of a $p$-variate elliptical distribution. We do so in challenging double asymptotic scenarios for which this direction eventually fails to be identifiable. In order to achieve robustness not only with respect to such weak identifiability but also with respect to heavy tails, we focus on sign-based statistical procedures, that is, on procedures that involve the observations only through their direction from the center of the distribution. We actually consider the generic problem of testing the null hypothesis that the first principal direction coincides with a given direction of $\mathbb{R}^{p}$. We first focus on weak identifiability setups involving single spikes (that is, involving spectra for which the smallest eigenvalue has multiplicity $p-1$). We show that, irrespective of the degree of weak identifiability, such setups offer local alternatives for which the corresponding sequence of statistical experiments converges in the Le Cam sense. Interestingly, the limiting experiments depend on the degree of weak identifiability. We exploit this convergence result to build optimal sign tests for the problem considered. In classical asymptotic scenarios where the spectrum is fixed, these tests are shown to be asymptotically equivalent to the sign-based likelihood ratio tests available in the literature. Unlike the latter, however, the proposed sign tests are robust to arbitrarily weak identifiability. We show that our tests meet the asymptotic level constraint irrespective of the structure of the spectrum, hence also in possibly multi-spike setups. We fully characterize the non-null asymptotic distributions of the corresponding test statistics under weak identifiability, which allows us to quantify the corresponding local asymptotic powers. Finally, Monte Carlo exercises are conducted to assess the finite-sample relevance of our asymptotic results and a real-data illustration is provided.

Citation

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Davy Paindaveine. Julien Remy. Thomas Verdebout. "Sign tests for weak principal directions." Bernoulli 26 (4) 2987 - 3016, November 2020. https://doi.org/10.3150/20-BEJ1213

Information

Received: 1 December 2018; Revised: 1 August 2019; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256166
MathSciNet: MR4140535
Digital Object Identifier: 10.3150/20-BEJ1213

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 4 • November 2020
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