Abstract.    The circular normal distribution, CN (mu, kappa), 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 kappa is known, and unknown respectively, the MLE \widehat mu of the mean direction mu is the best equivariant estimator. These results are generalized for the MLE \widehat {\displaystyle\mathop{mu}} of the mean direction vector \displaystyle\mathop {mu} = (mu₁,..., mu_p)^' in the simultaneous estimation problem with independent CN (mu_i, kappa), i = 1, ..., p, populations. We further observe that \widehat{\displaystyle\mathop {mu}} is admissible both when kappa 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(\displaystyle\mathop {mu}_i,kappa), i = 1,..., p.

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

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