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  "body": "\n## Definition\n\nLet $p \\ge 3$ and let $X$ have the [[multivariate normal distribution|0XN3K7RSZY]] on $\\R^p$ with unknown mean $\\theta \\in \\R^p$ and covariance matrix $\\sigma^2 I_p$, where $\\sigma^2 \u003e 0$ is known.\nThe *James–Stein estimator* of $\\theta$ is the [[estimator|0SSS6ERQ0S]]\n$$\\delta_{JS}(X) = \\Big( 1 - \\frac{(p - 2) \\sigma^2}{\\norm{X}^2} \\Big) X,$$\nwhere $\\norm{\\cdot}$ is the Euclidean norm.\n\n## Remarks\n\nThe estimator is defined with probability $1$, since $X \\ne 0$ almost surely.\nIt is a [[shrinkage estimator|GPBSS1V3SS]] toward $0$; shrinking toward a fixed point $\\nu \\in \\R^p$ gives $\\nu + \\big(1 - (p - 2)\\sigma^2 / \\norm{X - \\nu}^2\\big)(X - \\nu)$.\nWhen $\\norm{X}^2 \u003c (p - 2)\\sigma^2$ the factor is negative; the *positive-part* estimator replaces the factor by its positive part, and it has even smaller risk.\n\nUnder the loss $\\norm{a - \\theta}^2$, the estimators $(1 - c\\sigma^2/\\norm{X}^2) X$ have [[risk|N26TSVRH2S]] $p\\sigma^2 - (2c(p - 2) - c^2) \\sigma^4 E_\\theta[1/\\norm{X}^2]$.\nThis is below $p\\sigma^2$, the risk of $X$, exactly when $0 \u003c c \u003c 2(p - 2)$, and it is smallest for $c = p - 2$, the James–Stein choice.\n\nExample: $p = 3$, $\\sigma = 1$ and the observation $X = (1, 1, 1)$.\nThen $\\norm{X}^2 = 3$, the factor is $1 - 1/3 = 2/3$, and $\\delta_{JS}(X) = (2/3, 2/3, 2/3)$.\n\nNon-example: the same formula with $c = 2(p - 2)$ in place of $p - 2$.\nIt also shrinks toward $0$, but by the risk formula above its risk equals $p\\sigma^2$ for every $\\theta$, so it gives no improvement over $X$.\n",
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