James-Stein Estimator

The James-Stein estimator is an estimator of the unknown population mean parameter of a multivariate normal distribution that improves on the observation vector under expected total squared-error loss in dimension . For , where is the identity matrix, it is defined by

for , with . Here denotes the L²-norm of the vector (James and Stein 1961, pp. 363-365).

For the squared-error loss function , the risk, or expected loss, is

The displayed expectation value is finite and positive for . In comparison, the usual estimator has constant risk . At , the James-Stein risk is 2 (James and Stein 1961, p. 363).

The improvement concerns the expected sum of squared coordinate errors, for every . It need not hold for each coordinate separately or for each observed vector. Also, the multiplier is negative when . The formula therefore does not always place the estimate between the origin and the observation. The improvement over is the phenomenon known as Stein's paradox.