  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.
