High-dimensional tests for spherical location and spiked covariance

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  nn to be large compared to the dimension  pp. 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  nn and  pp go to infinity. This reveals that (i) the Watson test is robust against high dimensionality, and that (ii) it allows for (n,p)(n,p)-asymptotic results that are universal, in the sense that pp may go to infinity arbitrarily fast (or slowly) as a function of  nn. 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.