"On $m$-points uniformity tests on hyperspheres", accepted in Bernoulli

The paper “On $m$-points uniformity tests on hyperspheres”, coauthored by Alberto Fernández-de-Marcos, Eduardo García-Portugués, and Thomas Verdebout, has been accepted for publication in the renowned Bernoulli!

This is the abstract of the paper:

When testing for the uniformity of directions, most classical approaches fall within the class of Sobolev tests. In this work, we propose generalizations of Sobolev tests via two new families of uniformity tests based on $U$- and $V$-statistics featuring kernels of arbitrary degree $m$ that capture interactions among $m$-tuples of observations. Our tests encompass the classical Sobolev tests as a special case when $m=2$. We demonstrate that the computation of these new $V$-statistics remains tractable even for large degrees and sample sizes, and we provide closed-form expressions for circular $m$-points test statistics. We investigate the asymptotic behavior of our $m$-points statistics under the null hypothesis and obtain non-standard results involving random Hermite polynomials. We also derive their asymptotic properties under both fixed and local alternatives. Through simulations, we show that tests with $m>2$ yield important gains in power across several scenarios compared to classical Sobolev tests. Furthermore, we investigate the impact of $m$ on the rotational invariance and the effect of invariantization in the asymptotic null distributions.