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