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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a statistical model with a loss function, and let $R(\\theta, \\delta)$ be the [[risk function|N26TSVRH2S]] of an [[estimator|0SSS6ERQ0S]] $\\delta$.\nThe *minimax risk* of the problem is\n$$R^* = \\inf_{\\delta} \\sup_{\\theta \\in \\Theta} R(\\theta, \\delta),$$\nwhere the infimum is over all estimators.\nAn estimator $\\delta^*$ is *minimax* if\n$$\\sup_{\\theta \\in \\Theta} R(\\theta, \\delta^*) = R^*.$$\n\n## Remarks\n\nA minimax estimator need not exist, and it need not be unique.\nA Bayes estimator whose risk function is constant in $\\theta$ is minimax.\n\nExample: let $X$ be [[binomial|0WCWA1KT3M]] with parameters $n$ and $\\theta \\in [0, 1]$, with squared error loss.\nLet $\\delta^*(X) = (X + \\sqrt{n}/2)/(n + \\sqrt{n})$.\nA direct computation of variance plus squared bias gives $R(\\theta, \\delta^*) = \\frac{1}{4(1 + \\sqrt{n})^2}$ for every $\\theta$.\nThis $\\delta^*$ is the Bayes estimator for the beta prior with parameters $\\sqrt{n}/2$ and $\\sqrt{n}/2$, and its risk is constant, so it is minimax.\n\nNon-example: $\\delta(X) = X/n$ in the same problem.\nIts risk is $\\theta(1 - \\theta)/n$, with maximum $\\frac{1}{4n}$ at $\\theta = \\frac{1}{2}$.\nSince $\\frac{1}{4n} \u003e \\frac{1}{4(1 + \\sqrt{n})^2}$, its worst-case risk exceeds $R^*$, so it is not minimax.\n",
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