R.L. Mason and J.P. Keating
Southwest Research Institute and University of Texas at San Antonio

Stein's initial estimator of a multivariate normal mean

In the context of the multivariate normal distribution, Charles Stein (1956) presented a proof that in three or higher dimensions, the estimator of the population mean vector, based on components that are individually unbiased for corresponding components of the population mean vector, is not best in the sense of mean squared error. Stein presents an alternative estimator, which has uniformly smaller mean squared error than the vector whose components are unbiased estimators of corresponding components of the population mean vector. Within Stein's original paper, however, he suggests another estimator which he says did not shrink enough. We examine this initial estimator which we think may provide insight into how Professor Stein discovered the wonderful phenomenon known as "Stein's phenomenon."

Key words: Stein estimation, Hyperbolic geometry, Mean Squared Error