Abstract

We consider the problem of testing the null hypothesis that the direction theta of a skewed single-spiked high-dimensional distribution coincides with a given direction theta_0. For robustness purposes, we restrict to spatial sign tests, that is, to tests that involve the observations only through their projections onto the unit sphere. This reduces the problem to a classical problem in directional statistics, namely to the spherical location testing problem, for which the Watson test is the standard procedure. We study the asymptotic null and non-null behaviours of this test, in a general asymptotic framework where the dimension converges to infinity in an arbitrary way as a function of the sample size n. We also allow the strength of the signal to behave in a completely free way with n, which provides a complete spectrum of problems ranging from arbitrarily challenging to arbitrarily easy problems. Our results identify several asymptotic regimes leading to different limiting asymptotic experiments. Asymptotically optimal tests are obtained in each regime. Monte Carlo studies support our theoretical results.