Abstract
Testing uniformity on the $p$-dimensional unit sphere is arguably the most fundamental problem in directional statistics. In this paper, we consider this problem in the framework of axial data, that is, under the assumption that the $n$ observations at hand are randomly drawn from a distribution that charges antipodal regions equally. More precisely, we focus on axial, rotationally symmetric, alternatives and first address the problem under which the direction $\boldsymbol{\theta}$ of the corresponding symmetry axis is specified. In this setup, we obtain Le Cam optimal tests of uniformity, that are based on the sample covariance matrix (unlike their non-axial analogs, that are based on the sample average). For the more important unspecified-$\boldsymbol{\theta}$ problem, some classical tests are available in the literature, but virtually nothing is known on their non-null behavior. We therefore study the non-null behavior of the celebrated Bingham test and of other tests that exploit the single-spiked nature of the considered alternatives. We perform Monte Carlo exercises to investigate the finite-sample behavior of our tests and to show their agreement with our asymptotic results.
Highlights
Directional statistics are concerned with data taking values on the unit hypersphere Sp−1 := {x ∈ Rp : x 2 := x x = 1} of Rp
The most fundamental problem in directional statistics is the problem of testing for uniformity, which, for a random sample X1, . . . , Xn at hand, consists in testing the null hypothesis that the observations are sampled from the uniform distribution over Sp−1
Strong results have been obtained for non-axial tests of uniformity regarding their asymptotic power under suitable local alternatives to uniformity and even regarding their optimality, but virtually nothing is known in that direction for axial tests of uniformity
Summary
Strong results have been obtained for non-axial tests of uniformity regarding their asymptotic power under suitable local alternatives to uniformity and even regarding their optimality (we refer to Cutting, Paindaveine and Verdebout, 2017 and to the references therein), but virtually nothing is known in that direction for axial tests of uniformity This provides the main motivation for the present work, that intends to fill an important gap by studying the non-null behavior of some classical (and less classical) axial tests of uniformity. We will do so in the semiparametric distributional framework that has been classically considered for non-axial tests of uniformity, namely the framework of rotationally symmetric distributions indexed by a finite-dimensional parameter (κ, θ) ∈ R+ × Sp−1 and an infinite-dimensional parameter f ∈ F (a family of functions we define ) Within this semiparametric model, the null hypothesis of uniformity takes the form H0 : κ = 0.
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