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 $\mathcal{H}_{0} ; \boldsymbol{\theta} = \boldsymbol{\theta}_{0}$ under which the (rotational) symmetry center $\boldsymbol{\theta}$ is equal to a given value $\boldsymbol{\theta}_{0}$. 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 “$\boldsymbol{\theta}_{0}$-spiked” structure. Finally, Monte Carlo studies corroborate our asymptotic results and briefly explore non-null rejection frequencies.
