On Best Equivariance and Admissibility of Simultaneous MLE for Mean Direction Vectors of Several Langevin Distributions

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 $\widehat\mu$ of the mean direction μ is the best equivariant estimator. These results are generalized for the MLE $\widehat{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\mu } }$ of the mean direction vector $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\mu } = (\mu _1 , \ldots ,\mu _p )'$in the simultaneous estimation problem with independent CN(μ$_i$, ϰ), i = 1,..., p, populations. We further observe that $\widehat{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\mu } }$ 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($L(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\mu } _i ,\kappa ),i = 1, \ldots ,p$.