TESTING UNIFORMITY ON HIGH-DIMENSIONAL SPHERES AGAINST MONOTONE ROTATIONALLY SYMMETRIC ALTERNATIVES

• 成果类型: Article
• 署名作者: Cutting, Christine; Paindaveine, Davy; Verdebout, Thomas
• 署名单位: Universite Libre de Bruxelles; Universite Libre de Bruxelles
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1473
• 发表日期: 2017
• 页码: 1024-1058
• 关键词: nuisance parameter; distributions; sphericity; angles

摘要

We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in nonnull issues. We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for fixed modal location. and allows to derive locally asymptotically most powerful tests under specified.. The second one, that addresses the Fisher-von Mises-Langevin (FvML) case, relates to the unspecified-theta problem and shows that the high-dimensional Rayleigh test is locally asymptotically most powerful invariant. Under mild assumptions, we derive the asymptotic nonnull distribution of this test, which allows to extend away from the FvML case the asymptotic powers obtained there from Le Cam's third lemma. Throughout, we allow the dimension p to go to infinity in an arbitrary way as a function of the sample size n. Some of our results also strengthen the local optimality properties of the Rayleigh test in low dimensions. We perform aMonte Carlo study to illustrate our asymptotic results. Finally, we treat an application related to testing for sphericity in high dimensions.