*J. Japan Statist. Soc.,* Vol. 31 (No. 2), pp. 239-256, 2001

## Estimation of a Mean Vector Based on the Optimum Estimator Under the Known Norm Component

Teruo Fujioka and Takemi Yanagimoto

**Abstract. **
A unified treatment of the estimation of a mean vector in the normal and the inverse Gaussian distributions is discussed. A mean vector in the exponential dispersion model is reparametrized into two orthogonal components; the norm component and the direction. We point out first that the optimum (shrinkage) factor is obtained in an explicit form, when the norm component is known. Then several candidate estimators of a mean vector are discussed in relation with this optimum factor, when the norm component is unknown. The results in the case of the normal distribution provide us with a novel view of the James-Stein estimator and the positive-part Stein estimator. Parallel treatments are possible in estimating a mean vector in the inverse Gaussian case. Extensions to the gamma case are discussed to some extent.

*Key words and phrases*:
Estimation orthogonal, Inverse Gaussian distribution, James-Stein estimator, Modified Bessel function, Projection.

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