TESTS FOR A GIVEN LINEAR STRUCTURE OF THE MEAN DIRECTION OF THE LANGEVIN DISTRIBUTION

YOKO WATAMORI

Department of Mathematics, Faculty of Science, Hiroshima University, Naka-ku, Hiroshima 730, Japan

(Received March 9, 1990; revised November 5, 1990)

Abstract. This paper deals with Watson statistic T_{W} and likelihood ratio (LR) statistic T_{L} for testing hypothesis H_{0s} : μ μ μ μ μ μ μ μ ∈ V (a given s-dimensional subspace) based on a sample of size n from a p -variate Langevin distribution M_{p}(μ μ μ μ μ μ μ μ, κ). Asymptotic expansions of the null and non-null distributions of T_{W} and T_{L} are obtained when n is large. Asymptotic expressions of those powers are also obtained. It is shown that the powers of them are coincident up to the order n^{−1} when κ is unknown.

Key words and phrases: Asymptotic expansion, central limit theorem, Langevin distribution, likelihood ratio statistic, Watson statistic, power comparison.