J. Japan Statist. Soc., Vol. 33 (No. 1), pp. 39-64, 2003

Electrostatic views of Stein-type estimation of location vectors

Toshio Ohnishi and Takemi Yanagimoto

Abstract. Stein-type estimation of location vectors is discussed with the aid of the theory of electrostatics. We consider a class of estimating functions and assess the superiority of an estimating equation by its mean squared norm. The Coulomb potential function leads to a Pythagorean relationship with respect to this norm. By making full use of the Pythagorean relationship, we improve upon the likelihood estimating function. A further improvement is shown to be feasible under a certain condition which is described. We pursue possible strong relationships between the superiority over the likelihood estimating function and physical quantities appearing in the theory of electrostatics.

Key words and phrases: Coulomb potential function, electrostatics, estimating function, Green's formula, James-Stein estimator, James-Stein positive-part estimator, Pythagorean relationship.

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