High-dimensional tests for spherical location and spiked covariance
Abstract
Rotationally symmetric distributions on the p-dimensional unit hypersphere, extremely popular in directional statistics, involve a location parameter theta that indicates the direction of the symmetry axis. The most classical way of addressing the spherical location problem H_0:theta=theta_0, with theta_0 a fixed location, is the so-called Watson test, which is based on the sample mean of the observations. This test enjoys many desirable properties, but its implementation 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 this work, we therefore introduce a modified Watson statistic that can cope with high-dimensionality. We derive its asymptotic null distribution as both n and p go to infinity. This is achieved in a universal asymptotic framework that allows p to go to infinity arbitrarily fast (or slowly) as a function of n. We further show that our results also provide high-dimensional tests for a problem that has recently attracted much attention, namely that of testing that the covariance matrix of a multinormal distribution has a "theta_0-spiked" structure. Finally, a Monte Carlo simulation study corroborates our asymptotic results.
Cite
@article{arxiv.1402.2823,
title = {High-dimensional tests for spherical location and spiked covariance},
author = {Christophe Ley and Davy Paindaveine and Thomas Verdebout},
journal= {arXiv preprint arXiv:1402.2823},
year = {2014}
}