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  "body": "\n## Counterexample\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 the identity $I_p$.\nEstimate $\\theta$ under the loss $L(\\theta, a) = \\norm{a - \\theta}^2$, the squared Euclidean distance.\n\nThe estimator $\\delta_0(X) = X$ has [[risk|N26TSVRH2S]] $R(\\theta, \\delta_0) = E_\\theta \\norm{X - \\theta}^2 = p$ for every $\\theta$.\nIt is not [[admissible|DSTF6X8DRJ]].\nThe [[James–Stein estimator|709T196XCT]]\n$$\\delta_{JS}(X) = \\Big( 1 - \\frac{p - 2}{\\norm{X}^2} \\Big) X$$\nhas risk\n$$R(\\theta, \\delta_{JS}) = p - (p - 2)^2 \\, E_\\theta\\Big[ \\frac{1}{\\norm{X}^2} \\Big].$$\nFor $p \\ge 3$ the expectation is finite and positive, so $R(\\theta, \\delta_{JS}) \u003c p$ for every $\\theta$.\nThus $\\delta_{JS}$ dominates $X$.\n\nCheck of the risk formula: write $\\delta_{JS}(X) = X + g(X)$ with $g(x) = -(p - 2) x / \\norm{x}^2$.\nThen $\\norm{g(x)}^2 = (p - 2)^2 / \\norm{x}^2$ and $\\sum_i \\partial_i g_i(x) = -(p - 2)^2 / \\norm{x}^2$.\n[[Stein's unbiased risk estimate|30XVCMK916]] gives $R(\\theta, \\delta_{JS}) = E_\\theta\\big[p + \\norm{g(X)}^2 + 2 \\sum_i \\partial_i g_i(X)\\big]$, which is the formula above.\n\n## Remarks\n\nFor $p = 1$ and $p = 2$, the estimator $X$ is admissible, so the dimension bound is sharp.\nThe improvement is largest near $\\theta = 0$, the point toward which $\\delta_{JS}$ shrinks: there $E_0[1/\\norm{X}^2] = 1/(p - 2)$, so the risk is $2$ instead of $p$.\nAny fixed point can replace $0$ as the target of the shrinkage.\n\n## Explanations\n\n### L1 human\n\nTo estimate three or more unrelated averages at once, the obvious method is to use each observed value as its own estimate.\nStein showed that this obvious method can be beaten for every possible truth, by pulling all estimates a little toward a common point.\nThe total squared error goes down, even though the quantities have nothing to do with each other.\nThis surprising fact changed how statisticians think about estimating many parameters together.\n",
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