James–Stein estimator

For one observation $X\sim N_p(\theta ,I_p)$ with $p\ge 3$, the James–Stein estimator isUnder quadratic loss, Stein's risk identity givesso it dominates the usual estimator $X$.

For $X\sim N_p(\theta ,I)$ and $p\ge 4$, this estimator preserves the sample mean and applies James–Stein shrinkage in its orthogonal $(p-1)$-dimensional contrast subspace. Its risk under quadratic loss is $p-(p-3{)}^2{\mathbb{E}}_{\theta }\Vert X-\bar{X}{1}_p{\Vert }^{-2}$. The noncentrality is $\Vert \theta -\bar{\theta }{1}_p{\Vert }^2$, so the reciprocal moment of a noncentral chi-squared variable makes the risk minimal exactly when all true components coincide; the minimum is $3$. A negative realized shrinkage factor reverses contrast signs, and its magnitude can exceed one for sufficiently small contrast norm.

The James-Stein risk is strictly below the dimension at every finite parameter, but its improvement tends to zero as the mean norm tends to infinity. Strict pointwise domination therefore leaves the supremum risk unchanged.