## The Annals of Statistics

### Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 45, Number 3 (2017), 1024-1058.

Dates
Revised: April 2016
First available in Project Euclid: 13 June 2017

https://projecteuclid.org/euclid.aos/1497319687

Digital Object Identifier
doi:10.1214/16-AOS1473

Mathematical Reviews number (MathSciNet)
MR3662447

Zentralblatt MATH identifier
1368.62133

Subjects
Primary: 62H11: Directional data; spatial statistics 62G20: Asymptotic properties
Secondary: 62H15: Hypothesis testing

#### Citation

Cutting, Christine; Paindaveine, Davy; Verdebout, Thomas. Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives. Ann. Statist. 45 (2017), no. 3, 1024--1058. doi:10.1214/16-AOS1473. https://projecteuclid.org/euclid.aos/1497319687

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#### Supplemental materials

• Supplement to “Testing uniformity on high-dimensional spheres against rotationally symmetric alternatives”. In this supplementary article, we derive the fixed-$p$ asymptotic nonnull distribution of the Rayleigh test statistic in (3.4), and we show that, under FvML distributions, the conditions (i)–(iii) of Theorem 5.1 always hold.