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  "body": "\n## Definition\n\nLet $x_1, \\dots, x_n$ be data with $n \\ge 2$, and let $\\hat\\theta = T_n(x_1, \\dots, x_n)$ be a [[statistic|036756NFD6]] defined for every sample size.\nFor $i = 1, \\dots, n$, let $\\hat\\theta_{(i)} = T_{n-1}(x_1, \\dots, x_{i-1}, x_{i+1}, \\dots, x_n)$ be the leave-one-out value, and let $\\hat\\theta_{(\\cdot)} = \\frac{1}{n} \\sum_{i=1}^{n} \\hat\\theta_{(i)}$.\nThe *jackknife* gives:\n\n1. the bias estimate $\\widehat{\\mathrm{bias}} = (n - 1)\\big(\\hat\\theta_{(\\cdot)} - \\hat\\theta\\big)$;\n2. the bias-corrected estimate $\\hat\\theta_{\\mathrm{jack}} = \\hat\\theta - \\widehat{\\mathrm{bias}} = n\\hat\\theta - (n - 1)\\hat\\theta_{(\\cdot)}$;\n3. the variance estimate $\\widehat{\\operatorname{Var}}_{\\mathrm{jack}} = \\frac{n - 1}{n} \\sum_{i=1}^{n} \\big(\\hat\\theta_{(i)} - \\hat\\theta_{(\\cdot)}\\big)^2$.\n\n## Remarks\n\nIf the [[bias|01BD2TFFP4]] of $\\hat\\theta$ has the form $a/n + O(1/n^2)$, the bias of $\\hat\\theta_{\\mathrm{jack}}$ is $O(1/n^2)$.\n\nExample: for the sample mean $\\hat\\theta = \\bar x$, one has $\\hat\\theta_{(i)} = (n \\bar x - x_i)/(n - 1)$, so $\\hat\\theta_{(i)} - \\hat\\theta_{(\\cdot)} = (\\bar x - x_i)/(n-1)$.\nThen $\\widehat{\\operatorname{Var}}_{\\mathrm{jack}} = \\frac{1}{n(n-1)} \\sum_i (x_i - \\bar x)^2 = s^2/n$, the usual estimate of the [[variance|007QED3E0Y]] of $\\bar x$, with $s^2$ the [[sample variance|0JN9P4A944]].\nFor the plug-in variance $\\hat\\theta = \\frac1n \\sum_i (x_i - \\bar x)^2$, the bias-corrected estimate is exactly $s^2 = \\frac{1}{n-1}\\sum_i (x_i - \\bar x)^2$.\n\nNon-example: for the sample median the variance estimate in clause 3 is not consistent.\nThe median is not a smooth function of the data, and its leave-one-out values take only two or three distinct values.\n",
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