Detecting the direction of a signal on high-dimensional spheres: Non-null and Le Cam optimality results
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We consider one of the most important problems in directional statistics, namely the problem of testing the null hypothesis that the spike direction θ of a Fisher-von Mises-Langevin distribution on the p-dimensional unit hypersphere is equal to a given direction θ_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 κ_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 (κ_n), that yield different contiguity rates and different limiting experiments. In each regime, we derive Le Cam optimal tests under specified κ_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 κ_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.
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