High-dimensional tests for spherical location and spiked covariance

Christophe Ley, Davy Paindaveine and Thomas Verdebout

Journal of Multivariate Analysis, 2015, vol. 139, issue C, 79-91

Abstract: This paper mainly focuses on one of the most classical testing problems in directional statistics, namely the spherical location problem that consists in testing the null hypothesis H0:θ=θ0 under which the (rotational) symmetry center θ is equal to a given value θ0. The most classical procedure for this problem is the so-called Watson test, which is based on the sample mean of the observations. This test enjoys many desirable properties, but its asymptotic theory requires the sample size n to be large compared to the dimension p. This is a severe limitation, since more and more problems nowadays involve high-dimensional directional data (e.g., in genetics or text mining). In the present work, we derive the asymptotic null distribution of the Watson statistic as both n and p go to infinity. This reveals that (i) the Watson test is robust against high dimensionality, and that (ii) it allows for (n,p)-asymptotic results that are universal, in the sense that p may go to infinity arbitrarily fast (or slowly) as a function of n. Turning to Euclidean data, we show that our results also lead to a test for the null that the covariance matrix of a high-dimensional multinormal distribution has a “θ0-spiked” structure. Finally, Monte Carlo studies corroborate our asymptotic results and briefly explore non-null rejection frequencies.

Keywords: Directional statistics; High-dimensional data; Location tests; Principal component analysis; Rotationally symmetric distributions; Spherical mean (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (6)

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DOI: 10.1016/j.jmva.2015.02.019

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