= James–Stein shrinkage toward the sample mean
{c}
{title2=$\widehat\theta=\bar X1_p+(1-(p-3)/\|X-\bar X1_p\|^2)(X-\bar X1_p)$}
For $X\sim N_p(\theta,I)$ and $p\geq4$, 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 $p-(p-3)^2\mathbb E_\theta\|X-\bar X1_p\|^{-2}$. The noncentrality is $\|\theta-\bar\theta1_p\|^2$, 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>.
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