Information About

James-stein Estimator




An earlier version of the estimator was developed by Stein (1956), and is sometimes referred to as Stein's estimator. The result was improved by James and Stein (1961).


SETTING

Suppose \boldsymbol heta is an unknown parameter vector of length m, and let \mathbf y be observations of the parameter vector, such that

:
{\mathbf y} \sim N({\boldsymbol heta}, \sigma^2 I).\,


We are interested in obtaining an estimate \widehat{\boldsymbol heta} = \widehat{\boldsymbol heta}({\mathbf y}) of \boldsymbol heta, based on the observations \mathbf y.

This is an everyday situation in which a set of parameters is measured, and the measurements are corrupted by independent Gaussian noise. Since the noise has zero mean, it is very reasonable to use the measurements themselves as an estimate of the parameters. This is the approach of the Least Squares estimator, which simply equals \widehat{\boldsymbol heta}_{LS} = {\mathbf y} in this case.