Title: Detecting the direction of a signal on high-dimensional spheres: Non-null and Le Cam optimality results,
Abstract: We consider one of the most important problems in directional statistics,
namely the problem of testing the null hypothesis that the spike direction
${\pmb \theta}$ of a Fisher-von Mises-Langevin distribution on the
$p$-dimensional unit hypersphere is equal to a given direction ${\pmb
\theta}_0$. After a reduction through invariance arguments, we derive local
asymptotic normality (LAN) results in a general high-dimensional framework
where the dimension $p_n$ goes to infinity at an arbitrary rate with the sample
size $n$, and where the concentration $\kappa_n$ behaves in a completely free
way with $n$, which offers a spectrum of problems ranging from arbitrarily easy
to arbitrarily challenging ones. We identify seven asymptotic regimes,
depending on the convergence/divergence properties of $(\kappa_n)$, that yield
different contiguity rates and different limiting experiments. In each regime,
we derive Le Cam optimal tests under specified $\kappa_n$ and we compute, from
the Le Cam third lemma, asymptotic powers of the classical Watson test under
contiguous alternatives. We further establish LAN results with respect to both
spike direction and concentration, which allows us to discuss optimality also
under unspecified $\kappa_n$. To obtain a full understanding of the non-null
behavior of the Watson test, we use martingale CLTs to derive its local
asymptotic powers in the broader, semiparametric, model of rotationally
symmetric distributions. A Monte Carlo study shows that the finite-sample
behaviors of the various tests remarkably agree with our asymptotic results.