ON BEST EQUIVARIANCE AND ADMISSIBILITY OF SIMULTANEOUS MLE FOR MEAN DIRECTION VECTORS OF

SEVERAL LANGEVIN DISTRIBUTIONS

ASHIS SENGUPTA^{1} AND RANJAN MAITRA^{2}

1Computer Science Unit, Indian Statistical Institute, Calcutta, India and Department of Statistics and AP, University of California, Santa Barbara, U.S.A.

2Department of Statistics, University of Washington, Seattle, U.S.A.

(Received September 30, 1993; revised March 26, 1997)

Abstract. The circular normal distribution, CN(μ, κ), plays a role for angular data comparable to that of a normal distribution for linear data. We establish that for the curved and for the regular exponential family situations arising when κ is known, and unknown respectively, the MLE μ̂ of the mean direction μ is the best equivariant estimator. These results are generalized for the MLE ˆ μ_{∼} of the mean direction vector μ_{∼} = (μ_{1}, …, μ_{p})′ in the simultaneous estimation problem with independent CN(μ_{i}, κ), i = 1, …, p, populations. We further observe that ˆ μ_{∼} is admissible both when κ is known or unknown. Thus unlike the normal theory, Stein effect does not hold for the circular normal case. This result is generalized for the simultaneous estimation problem with directional data in q-dimensional hyperspheres following independent Langevin distributions, L(μ_{∼}_{i}, κ), i = 1, …, p.

Key words and phrases: Admissibility of estimators, Bayes estimators, best equivariant estimator, Langevin distribution, mean direction vector, Stein effect.