On the power of Sobolev tests for isotropy under local rotationally symmetric alternatives

Abstract

We consider one of the most classical problems in multivariate statistics, namely the problem of testing isotropy, or equivalently, the problem of testing uniformity on the unit hypersphere S p−1 of R p . Rather than restricting to tests that can detect specific types of alternatives only, we consider the broad class of Sobolev tests. While these tests are known to allow for omnibus testing of uniformity, their non-null behavior and consistency rates, unexpectedly, remain largely unexplored. To improve on this, we thoroughly study the local asymptotic powers of Sobolev tests under the most classical alternatives to uniformity, namely, under rotationally symmetric alternatives. We show in particular that the consistency rate of Sobolev tests does not only depend on the coefficients defining these tests but also on the derivatives of the underlying angular function at zero. For any Sobolev test and any rotationally symmetric alternative, we derive the consistency rate of the Sobolev test and determine the corresponding local asymptotic powers. We show that Sobolev tests with non-zero coefficients at odd (respectively, even) ranks only are blind (at any polynomial rate) to alternatives with angular functions whose kth-order derivatives at zero vanish for any k odd (respectively, even). Our asymptotic analysis requires investigating the non-standard behavior of random Chebyshev polynomials (for p = 2) and random Gegenbauer polynomials (for p ≥ 3) in the vicinity of the uniform distribution on S p−1 . Our non-standard asymptotic results are illustrated with Monte Carlo exercises.

A general maximal projection approach to uniformity testing on the hypersphere

“…Essentially, this is because the local Bahadur efficiency does not make a statement about rates of consistency under which a statistic might nevertheless recognize the alternative. For example, the authors of [18] prove that the Bingham test, which has structural resemblance with T n, 2 , reliably recognizes a von Mises-Fisher distribution under certain rates of consistency. Hence, it might be interesting to explore the problem of rates of consistency, possibly even for very general alternatives like rotationally symmetric distributions.…”

“…Another indication for the inconsistency of T n, β can be seen in the case β = 1, which essentially concerns the Rayleigh test. The authors of [18] have shown that, in the rather general context of rotationally symmetric alternatives with a location and concentration parameter and a defining angular function, the Rayleigh test is blind against certain local alternatives. These local alternatives show polynomial decrease of the concentration parameter towards zero (hence yielding the null hypothesis) and the odd-order derivatives of their angular function vanish at zero.…”