*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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