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  "body": "\n## Definition\n\nLet $X$ be an observation from a statistical model with parameter $\\theta \\in \\R^p$.\nLet $\\delta_0(X)$ be an [[estimator|0SSS6ERQ0S]] of $\\theta$ and let $\\nu \\in \\R^p$ be a fixed target.\nLet $w(X)$ be a [[statistic|0Q5CG2NEQ8]] with values in $[0, 1]$.\nThe estimator\n$$\\delta(X) = \\nu + w(X) \\big( \\delta_0(X) - \\nu \\big)$$\nis a *shrinkage estimator* of $\\theta$, obtained by shrinking $\\delta_0$ toward $\\nu$.\n\n## Remarks\n\nThe weight $w$ may be a constant or may depend on the data.\nThe term is also used more broadly for any estimator that pulls a basic estimate toward a target, for example by a penalty in a regression.\nThe plain James–Stein estimator has a weight that can be negative; its positive-part version fits the definition above.\n\nExample: let $X$ be [[normal|05AQE0QR0D]] with mean $\\theta$ and known variance $\\sigma^2$, and give $\\theta$ the normal prior with mean $0$ and variance $\\tau^2$.\nThe [[posterior|0KJ7ZS8AJA]] mean is $\\frac{\\tau^2}{\\tau^2 + \\sigma^2} X$.\nThis is a shrinkage estimator with $\\delta_0 = X$, $\\nu = 0$ and constant weight $w = \\tau^2/(\\tau^2 + \\sigma^2) \\in (0, 1)$.\n\nNon-example: $\\delta(X) = 2X$ in the same model.\nIt has the form above with $\\nu = 0$ and $w = 2$, which is not in $[0, 1]$, so it pushes the estimate away from the target rather than toward it.\n",
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