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  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|JZ6R7J36K5]] and let $V = V(X)$ be a [[statistic|036756NFD6]] with values in a measurable space $(\\mathcal{V}, \\mathcal{B})$.\nThe statistic $V$ is *ancillary* if its distribution does not depend on $\\theta$: for every $B \\in \\mathcal{B}$, the probability $P_\\theta(V \\in B)$ is the same for all $\\theta \\in \\Theta$.\n\n## Remarks\n\nExample: let $X_1, \\dots, X_n$ be i.i.d. $\\mathcal{N}(\\mu, 1)$ with $\\mu \\in \\R$ and $n \\ge 2$.\nThen $X_i - \\mu$ are i.i.d. $\\mathcal{N}(0, 1)$, and the sample variance $S^2 = \\frac{1}{n-1}\\sum_i (X_i - \\bar{X})^2$ is a function of them alone, because $X_i - \\bar{X} = (X_i - \\mu) - (\\bar{X} - \\mu)$; so $S^2$ is ancillary for $\\mu$.\nNon-example: in the same model, $\\bar{X} \\sim \\mathcal{N}(\\mu, 1/n)$, whose law changes with $\\mu$, so $\\bar{X}$ is not ancillary.\nIn a location model $X_i = \\mu + \\varepsilon_i$ with a fixed law of the $\\varepsilon_i$, every function of the differences $X_i - X_j$ is ancillary.\nIn a scale model $X_i = \\sigma \\varepsilon_i$, every function of the ratios $X_i / X_j$ is ancillary, when these are defined.\n",
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    "name": "Ancillary statistic",
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  "msc": [
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  "name": "Ancillary statistic",
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