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SOME MODIFICATIONS OF IMPROVED ESTIMATORS

OF A NORMAL VARIANCE

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NOBUO SHINOZAKI

*Department of Administration Engineering, Faculty of
Science and Technology,*

Keio University, 3-14-1 Hiyoshi, Kohoku, Yokohama 223, Japan
(Received May 12, 1994; revised November 24, 1994)

**Abstract.**
Consider the problem of estimating a normal
variance based on a random sample when the mean is unknown. Scale
equivariant estimators which improve upon the best scale and
translation equivariant one have been proposed by several authors for
various loss functions including quadratic loss. However, at least
for quadratic loss function, improvement is not much. Herein, some
methods are proposed to construct improving estimators which are not
scale equivariant and are expected to be considerably better when the
true variance value is close to the specified one. The idea behind
the methods is to modify improving equivariant shrinkage estimators,
so that the resulting ones shrink little when the usual estimate is
less than the specified value and shrink much more otherwise.
Sufficient conditions are given for the estimators to dominate the
best scale and translation equivariant rule under the quadratic loss
and the entropy loss. Further, some results of a Monte Carlo
experiment are reported which show the significant improvements by
the proposed estimators.

*Key words and phrases*:
Entropy loss, quadratic loss,
shrinkage estimator, Stein estimator, uniform risk improvement.

**Source**
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