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ON BEST EQUIVARIANCE AND ADMISSIBILITY OF

SIMULTANEOUS MLE FOR MEAN DIRECTION VECTORS OF

SEVERAL LANGEVIN DISTRIBUTIONS

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

^{1} *Computer Science Unit, Indian Statistical Institute, Calcutta, India and*

Department of Statistics and AP, University of California, Santa Barbara, U.S.A.

^{2} *Department of Statistics, University of Washington, Seattle, U.S.A.*
(Received September 30, 1993; revised March 26, 1997)

**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*} = (*mu*_{1},..., *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.

**Source**
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