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TESTS FOR A GIVEN LINEAR STRUCTURE OF THE MEAN

DIRECTION OF THE LANGEVIN DISTRIBUTION

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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} : *mu* \in *V*
(a given *s*-dimensional subspace) based on a sample of size
*n* from a *p*-variate Langevin distribution *M*_{p}(*mu*,*kappa*).
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 *kappa* is unknown.

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

**Source**
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