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

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 $\mathcal{S}^{p-1}$ of $\mathbb{R}^p$. 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 $k$th-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 $p\geq 3$) in the vicinity of the uniform distribution on $\mathcal{S}^{p-1}$. Our non-standard asymptotic results are illustrated with Monte Carlo exercises.