A0879
Title: Testing uniformity on the sphere: From pairs to $m$-tuples
Authors: Alberto Fernandez de Marcos - Universidad Carlos III Madrid (Spain) [presenting]
Eduardo Garcia-Portugues - Universidad Carlos III de Madrid (Spain)
Abstract: When uniformity on the sphere is tested, most classical tests employ a $V$-statistic of second-degree investigating the relation between pairwise observations to detect deviations from uniformity. Two new classes of uniformity tests are proposed based on $U$- and $V$-statistics of arbitrary degree $m$ that capture the interaction between $m$-tuples of observations. It is proven that they extend the well-known Sobolev class of uniformity tests, arising for $m=2$. Through simulations, it is shown that tests with $m>2$ lead to increases in power, compared to $m=2$, for certain scenarios. The computation of the new class of tests is shown to be manageable even for large $m$ values, and closed-form circular test statistics are provided that extend classical tests for $m=2$. The asymptotic null distribution is studied for these $m$-tests, its usability in practice, and its asymptotic behavior under local alternatives. Furthermore, the impact of $m$ is investigated on the rotational invariance of the proposed test statistics, and a modified construction is introduced that ensures this invariance for all $m>2$, with closed expressions and known asymptotic distribution in the circular case.
