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Abstracts Session 1

9H30-10H45 SESSION 1: MATHEMATICAL STATISTICS

Laura Peralvo (ULB) and Thomas Verdebout (ULB): Testing for PCA under weak identifiability

Abstract: 

In this work, we study the asymptotic behavior of several tests for the eigenvector associated with the leading eigenvalue under weak identifiability. The tests we compare include signed-rank tests. We show that the latter are robust to weak identifiability and enjoy nice asymptotic power properties with respect to classical Gaussian competitors. Our results are illustrated via Monte-Carlo simulations.

 

Gaspard Bernard (ULB): 

Abstract: 

We tackle the problem of testing ${\cal H}_{0q}\n:  \lambda_{q,\Sigb}\n > \lambda_{q+1,\Sigb}\n= \ldots= \lambda_{p,\Sigb}\n$, where $\lambda_{1,\Sigb}\n, \ldots, \lambda_{p,\Sigb}\n$ are the ordered eigenvalues of scatter matrices $\Sigb\n$ of elliptical observations. We study robust procedures based on the multivariate signs of the observations. We show that the considered multivariate sign tests show very low power against alternatives of the type ${\cal H}_{1q}\n:  \lambda_{q,\Sigb}\n = \lambda_{q+1,\Sigb}\n= \ldots= \lambda_{p,\Sigb}\n$. We propose a new multivariate sign test showing arbitrarily large power in scenarios of the type ${\cal H}_{1q}\n$. Moreover, we show that this new multivariate sign test enjoys the same asymptotic local powers as the classical multivariate sign test of Bernard (2023) when $\lambda_{q,\Sigb}\n$ and $\lambda_{q+1,\Sigb}\n$ are sufficiently separated. We use this new test to construct a new robust estimator of the dimension of the signal.