Local Powers of Optimal One- and Multi-Sample Tests for the Concentration of Fisher-von Mises Langevin Distributions

One-sample and multi-sample tests on the concentration parameter of Fisher-vonMises-Langevin (FvML) distributions have been well studied in the literature. However,only very little is known about their behavior under local alternatives, whichis due to complications inherent to the curved nature of the parameter space. Theaim of the present paper therefore consists in filling that gap by having recourse tothe Le Cam methodology, which has been adapted from the linear to the sphericalsetup in Ley et al. (2013a). We obtain explicit expressions of the powers for the mostefficient one- and multi-sample tests; these tests are those considered in Watamori andJupp (2005). As a nice by-product, we are also able to write down the powers (againstlocal FvML alternatives) of the celebrated Rayleigh (1919) test of uniformity. A MonteCarlo simulation study confirms our theoretical findings and shows the finite-samplebehavior of the above-mentioned procedures.