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