Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion (TL(s)), Watson statistic (TW(s)) and Rao statistic (TR(s)) for testing H0s: [mu] [set membership, variant] (a given subspace) against H1s: [mu] [negated set membership] , based on a sample of size n from a p-variate Langevin distribution Mp([mu], ?) when ? is large. For the case when ? is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order ?-1. For the case when ? is unknown, it is shown that TR(s) [greater, double equals] TL(s) [greater, double equals] TW(s) in their powers up to the order ?-1.