2001 Volume 31 Issue 2 Pages 239-256
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.
