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

•Rank-based test for the concentration of directional data.•Asymptotic properties of a Kruskal-Wallis type test.•Asymptotic relative efficiency in the FvM case.

Recently, Verdebout (2015) introduced a Kruskal-Wallis type rank-based procedure ϕV(n) to test the homogeneity of concentrations of some distributions on the unit hypersphere Sp−1 of Rp. While the asymptotic properties of ϕV(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 ϕV(n) with respect to the optimal Fisher-von Mises (FvM) score test ϕWJ(n) of Watamori and Jupp (2005) in the FvM case. Quite surprisingly we obtain that in the vicinity of the uniform distribution of S2, ϕV(n) and ϕWJ(n) do perform almost equally well. This implies that the natural robustness of ϕV(n) that comes from the use of ranks has no asymptotic efficiency cost in the vicinity of the 3-dimensional uniform distribution.