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  "body": "\n## Statement\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[parametric model|MSC0MR8VMH]] for an observation $X$, and let $g \\colon \\Theta \\to \\R$.\nLet $T = T(X)$ be a [[sufficient statistic|WP4TGXJQJG]] that is [[complete|DED6JFAV1K]].\nLet $h$ be a measurable function such that $h(T)$ is [[unbiased|0BWGTN92FG]] for $g(\\theta)$ and $\\operatorname{Var}_\\theta(h(T)) \u003c \\infty$ for every $\\theta \\in \\Theta$.\nThen $h(T)$ is a [[uniformly minimum-variance unbiased estimator|ZCJX7THBNJ]] of $g(\\theta)$: for every unbiased estimator $\\delta$ of $g(\\theta)$ and every $\\theta \\in \\Theta$,\n$$\\operatorname{Var}_\\theta(h(T)) \\le \\operatorname{Var}_\\theta(\\delta).$$\nMoreover, $h(T)$ is the only such estimator, up to equality with $P_\\theta$-probability $1$ for every $\\theta$.\n\n## Remarks\n\nCompleteness gives uniqueness: if $h_1(T)$ and $h_2(T)$ are both unbiased for $g(\\theta)$, then $E_\\theta[h_1(T) - h_2(T)] = 0$ for all $\\theta$, so $h_1(T) = h_2(T)$ almost surely.\nGiven any unbiased $\\delta$ with finite [[variance|0477R9PYAE]], the [[Rao–Blackwell theorem|Y3TX4J41TP]] gives the unbiased estimator $E[\\delta \\mid T]$, a function of $T$, with variance at most $\\operatorname{Var}_\\theta(\\delta)$.\nBy uniqueness this function is $h(T)$, which proves the claim.\n\nExample: let $X_1, \\dots, X_n$ be independent Poisson variables with mean $\\lambda \u003e 0$.\nThe sum $T = X_1 + \\cdots + X_n$ is complete and sufficient, and $T/n$ is unbiased for $\\lambda$.\nSo the sample mean is the unique minimum-variance unbiased estimator of $\\lambda$.\n\nCompleteness cannot be dropped from the uniqueness argument.\nIf $T$ is not complete, two different functions of $T$ can be unbiased for the same $g(\\theta)$.\nThen the Rao–Blackwell step alone does not single out a best one.\n\n## Explanations\n\n### L1 human\n\nIf a summary of the data is sufficient and complete, then any unbiased estimate built only from that summary is automatically the best unbiased estimate.\n\"Best\" means smallest variance, for every value of the unknown parameter at once.\n\n### L2 human\n\nThis is the standard route to minimum-variance unbiased estimators.\nFirst find a complete sufficient statistic, often the natural statistic of an exponential family of full rank.\nThen find any function of it that is unbiased, for example by solving $E_\\theta[h(T)] = g(\\theta)$ or by Rao–Blackwellizing a simple unbiased estimator.\nThe theorem then certifies optimality without comparing with other estimators one by one.\n",
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