Stein's paradox is the phenomenon that estimating three or more population means jointly can reduce expected total squared-error loss, even for independent
observations from a normal distribution. Suppose
, ..., 
are independent
random variables with 
and 
. The goal is to estimate the vector 
under the
loss function 
. The usual estimator 
has risk, or expected loss, 
for every 
(Stein 1956, p. 197).
The James-Stein estimator has strictly smaller risk for every 
(James and Stein 1961, pp. 363-365). The independence of the observations does
not require an optimal joint rule to act separately on each coordinate.
Allowing estimator bias can reduce the expected
sum of squared errors, even though no improvement is guaranteed
for each coordinate or each observed data vector
(Numberphile 2025).
