On the efficiency of some rank-based test for the homogeneity of concentrations

Abstract

Recently, Verdebout (2015) introduced a Kruskal-Wallis type rank-based procedure ${\varphi }_V^{(n)}$ to test the homogeneity of concentrations of some distributions on the unit hypersphere ${\mathcal{S}}^{p-1}$ of ${\mathbb{R}}^p$. While the asymptotic properties of ${\varphi }_V^{(n)}$ are known under the null hypothesis, nothing is known about its behavior under local alternatives. In this paper we compute the asymptotic relative efficiency of phi((n))(V) with respect to the optimal Fisher-von Mises (FvM) score test ${\varphi }_{WJ}^{(n)}$ of Watamori and Jupp (2005) in the FvM case. Quite surprisingly we obtain that in the vicinity of the uniform distribution of ${\mathcal{S}}^2$, ${\varphi }_V^{(n)}$ and ${\varphi }_{WJ}^{(n)}$ do perform almost equally well. This implies that the natural robustness of ${\varphi }_V^{(n)}$ that comes from the use of ranks has no asymptotic efficiency cost in the vicinity of the 3-dimensional uniform distribution.