Detecting the direction of a signal on high-dimensional spheres: Non-null and Le Cam optimality results

TL;DR

This paper investigates the problem of testing the direction of a signal on high-dimensional spheres, deriving optimal tests and asymptotic powers across various regimes of dimension, concentration, and sample size.

Contribution

It provides a comprehensive high-dimensional asymptotic analysis of directional testing, including Le Cam optimality results and extensions to semiparametric models.

Findings

Derives local asymptotic normality results for high-dimensional directional data.

Identifies regimes with different contiguity rates and limiting experiments.

Establishes the asymptotic power of classical tests like Watson under various conditions.

Abstract

We consider one of the most important problems in directional statistics, namely the problem of testing the null hypothesis that the spike direction $\theta$ of a Fisher-von Mises-Langevin distribution on the $p$-dimensional unit hypersphere is equal to a given direction $\theta_0$. 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 various 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 investigate the non-null behavior of the Watson test outside the parametric framework above, we derive its local asymptotic powers through martingale CLTs 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.