SOME MODIFICATIONS OF IMPROVED ESTIMATORS OF A NORMAL VARIANCE

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.