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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] with an unknown distribution $F$, and let $\\hat\\theta = T(X_1, \\dots, X_n)$ be a [[statistic|036756NFD6]].\nLet $\\hat F_n$ be the [[empirical distribution|A05WR6TEK5]] of the observed values $x_1, \\dots, x_n$, which puts mass $1/n$ at each $x_i$.\nA *bootstrap sample* is a sample $X_1^*, \\dots, X_n^*$ i.i.d. from $\\hat F_n$, that is, $n$ draws with replacement from $x_1, \\dots, x_n$.\nThe *bootstrap distribution* of $\\hat\\theta$ is the law of $\\hat\\theta^* = T(X_1^*, \\dots, X_n^*)$ given the data.\nThe *bootstrap* estimates a feature of the [[sampling distribution|656Y8FA3RY]] of $\\hat\\theta$ under $F$ by the same feature of the bootstrap distribution.\nAn example of such a feature is the [[standard deviation|0VTN5RKNC5]].\n\n## Remarks\n\nIn practice one draws $B$ independent bootstrap samples, computes $\\hat\\theta^*_1, \\dots, \\hat\\theta^*_B$, and estimates the standard error of $\\hat\\theta$ by\n$$\\widehat{\\mathrm{se}}_B = \\Big( \\frac{1}{B - 1} \\sum_{b=1}^{B} \\big(\\hat\\theta^*_b - \\bar\\theta^*\\big)^2 \\Big)^{1/2}, \\qquad \\bar\\theta^* = \\frac{1}{B} \\sum_{b=1}^{B} \\hat\\theta^*_b.$$\nThe percentile interval takes the $\\alpha/2$ and $1 - \\alpha/2$ empirical quantiles of the $\\hat\\theta^*_b$ as a confidence interval of level about $1 - \\alpha$.\n\nExample: data $(1, 3)$ and $\\hat\\theta = \\bar X$.\nThe four equally likely bootstrap samples $(1,1), (1,3), (3,1), (3,3)$ have means $1, 2, 2, 3$.\nTheir variance is $(1 + 0 + 0 + 1)/4 = 1/2$, which equals $\\hat\\sigma^2 / n$ with the plug-in variance $\\hat\\sigma^2 = 1$.\n\nNon-example: for $X_i$ uniform on $[0, \\theta]$ and $\\hat\\theta = \\max_i X_i$, the bootstrap fails.\nA bootstrap sample repeats the observed maximum with probability $1 - (1 - 1/n)^n \\to 1 - e^{-1}$, so the bootstrap law of $n(\\hat\\theta - \\hat\\theta^*)$ has an atom at $0$, while the true law of $n(\\theta - \\hat\\theta)$ is continuous.\n\nThe [[jackknife|WQ8TQKDKYT]] is an older resampling method that leaves out one observation at a time.\n",
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