Open Access
June 2017 Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives
Christine Cutting, Davy Paindaveine, Thomas Verdebout
Ann. Statist. 45(3): 1024-1058 (June 2017). DOI: 10.1214/16-AOS1473


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 $\mathbf{{\theta}}$ and allows to derive locally asymptotically most powerful tests under specified $\mathbf{{\theta}}$. The second one, that addresses the Fisher–von Mises–Langevin (FvML) case, relates to the unspecified-$\mathbf{{\theta}}$ 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 a Monte Carlo study to illustrate our asymptotic results. Finally, we treat an application related to testing for sphericity in high dimensions.


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Christine Cutting. Davy Paindaveine. Thomas Verdebout. "Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives." Ann. Statist. 45 (3) 1024 - 1058, June 2017.


Received: 1 July 2015; Revised: 1 April 2016; Published: June 2017
First available in Project Euclid: 13 June 2017

zbMATH: 1368.62133
MathSciNet: MR3662447
Digital Object Identifier: 10.1214/16-AOS1473

Primary: 62G20 , 62H11
Secondary: 62H15

Keywords: contiguity , directional statistics , High-dimensional statistics , Invariance , local asymptotic normality , rotationally symmetric distributions , tests of uniformity

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 3 • June 2017
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