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

• 成果类型: Article
• 署名作者: Paindaveine, Davy; Verdebout, Thomas
• 署名单位: Universite Libre de Bruxelles; Universite Libre de Bruxelles; Universite de Toulouse; Universite Toulouse 1 Capitole; Toulouse School of Economics
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00937-x
• 发表日期: 2020
• 页码: 1165-1216
• 关键词: fisher; location; tests; distributions; regression; bessel

摘要

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 pn 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.