Spherically Symmetric Distributions

Abstract

In the previous chapters, estimation problems were considered for the normal distribution setting. Stein (1956) showed that the usual estimator of a location vector could be improved upon quite generally for p ≥ 3 and Brown (1966) substantially extended this conclusion to essentially arbitrary loss functions. Explicit results of the James-Stein type, however, have thus far been restricted to the case of the normal distribution. Recall the geometrical insight from Sect. 2.2.2, the development did not depend on the normality of X or even that θ is a location vector – this suggests that the improvement for Stein-type estimators may hold for more general distributions. Strawderman (1974a) first explored such an extension and considered estimation of the location parameter for scale mixtures of multivariate normal distributions.