 James–Stein shrinkage toward the sample mean

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.