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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a family of distributions of the data $X$, and let $g \\colon \\Theta \\to \\R$ be the quantity to estimate.\nAn [[estimator|0TTNRP5PFQ]] $T = T(X)$ is *unbiased* for $g$ if $E_\\theta(T) = g(\\theta)$ for every $\\theta \\in \\Theta$, that is, its [[bias|01BD2TFFP4]] is zero.\nLet $T$ be an unbiased estimator of $g$ with $\\operatorname{Var}_\\theta(T) \u003c \\infty$ for all $\\theta$.\nIt is a *minimum-variance unbiased estimator*, or *uniformly minimum-variance unbiased estimator* (UMVUE), if\n$$\\operatorname{Var}_\\theta(T) \\le \\operatorname{Var}_\\theta(T')$$\nfor every unbiased estimator $T'$ of $g$ and every $\\theta \\in \\Theta$.\n\n## Remarks\n\nHere $\\operatorname{Var}_\\theta$ is the [[variance|007QED3E0Y]] when $X \\sim P_\\theta$.\n\nExample: let $X_1, \\dots, X_n$ be i.i.d. [[Bernoulli|0J97P5KZY0]]($p$) with $\\Theta = (0, 1)$.\nThe proportion $\\bar{X}$ is unbiased with variance $p(1-p)/n$.\nThe Fisher information of the sample is $n/(p(1-p))$, so the Cramér–Rao bound for unbiased estimators of $p$ is $p(1-p)/n$.\nThe estimator $\\bar{X}$ attains it at every $p$, so $\\bar{X}$ is the UMVUE of $p$.\n\nNon-example: for $n \\ge 3$, $T' = (X_1 + X_2)/2$ is unbiased, but its variance $p(1-p)/2$ exceeds $p(1-p)/n$ for every $p$, so it fails the variance inequality.\n\nThe UMVUE need not exist, and an estimator that attains the Cramér–Rao bound need not exist even when the UMVUE does.\nIf $S$ is a complete sufficient statistic and $T$ is unbiased, then $E(T \\mid S)$ is the UMVUE; this is the Lehmann–Scheffé theorem.\n",
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