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One-sample and multi-sample tests on the concentration parameter ofFisher-von Mises-Langevin (FvML) distributions have been well studied in theliterature. However, only very little is known about their behavior under localalternatives, which is due to complications inherent to the curved nature ofthe parameter space. The aim of the present paper therefore consists in fillingthat gap by having recourse to the Le Cam methodology, which has been adaptedfrom the linear to the spherical setup in Ley \emph{et al.} (2013). We obtainexplicit expressions of the powers for the most efficient one- and multi-sampletests; these tests are those considered in Watamori and Jupp (2005). As a niceby-product, we are also able to write down the powers (against local FvMLalternatives) of the celebrated Rayleigh (1919) test of uniformity. A MonteCarlo simulation study confirms our theoretical findings and shows thefinite-sample behavior of the above-mentioned procedures.

Local powers of optimal one- and multi-sample tests for the concentration of Fisher-von Mises-Langevin distributions