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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|JZ6R7J36K5]] and let $T = T(X)$ be a [[statistic|036756NFD6]] with values in a measurable space $\\mathcal{T}$.\nThe statistic $T$ is *complete* if the following holds for every measurable $g \\colon \\mathcal{T} \\to \\R$ such that $g(T)$ is integrable under every $P_\\theta$.\nThe condition\n$$E_\\theta[g(T)] = 0 \\quad \\text{for all } \\theta \\in \\Theta$$\nimplies $P_\\theta(g(T) = 0) = 1$ for all $\\theta \\in \\Theta$.\nHere $E_\\theta$ is the [[expected value|0DCT0QZVQ1]] under $P_\\theta$.\nThe statistic is *boundedly complete* if this implication holds for every bounded measurable $g$.\n\n## Remarks\n\nEvery complete statistic is boundedly complete.\nExample: let $T$ have the binomial law with parameters $n$ and $p \\in (0, 1)$.\nIf $E_p[g(T)] = (1 - p)^n \\sum_{t=0}^{n} g(t)\\binom{n}{t} r^t = 0$ with $r = p/(1 - p)$ for all $p$, the polynomial in $r$ vanishes on $(0, \\infty)$, so each $g(t) = 0$; thus $T$ is complete.\nNon-example: let $X_1, X_2$ be i.i.d. $\\mathcal{N}(\\theta, 1)$ and $T = (X_1, X_2)$.\nThe function $g(T) = X_1 - X_2$ has $E_\\theta[g(T)] = 0$ for all $\\theta$, but $g(T) \\ne 0$ with probability $1$; so $T$ is not complete.\nCompleteness is a property of the family of laws $\\set{P_\\theta \\circ T^{-1} : \\theta \\in \\Theta}$; enlarging $\\Theta$ can only help it.\n",
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  "markdown": "https://tamnd.github.io/math2/node/PF4RHTZWXH.md",
  "msc": [
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  "name": "Complete statistic",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/PF4/PF4RHTZWXH.md",
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