On Sobolev tests of uniformity on the circle with an extension to the sphere (Q2174999)

Directional statistics is a branch of statistics that deals with observations that are directions or, more generally, observations lying on nonlinear manifolds. The authors provide a new test \(\phi_{MRV}^{(n)}\) on the circle. The test \(\phi_{MRV}^{(n)}\) is based on a standardized version of the quantity \(S_{1,M_n}\). The authors consider as test statistic in the standardized version \[ S_{1,M_n}^{\mathrm{stand}} := \frac{ S_{1,M_n} -M_n}{ 2 \sqrt{M_n}} \] of \(S_{1,M_n}\) for which the following result provides the asymptotic null distribution as \(M_n \rightarrow +\infty\) with \(M_n = o(n^2)\) as \(n \rightarrow +\infty\). In this paper the empirical power of the proposed test with several competitors is examined. The topic of the paper is rather interesting.