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  "body": "\n## Definition\n\nLet $T$ be an [[estimator|0TTNRP5PFQ]] that maps each sample $x = (x_1, \\dots, x_n)$ of points in $\\R^p$ to a value $T(x) \\in \\R^k$.\nFor $0 \\le m \\le n$, let $\\mathcal{Y}_m(x)$ be the set of samples obtained from $x$ by replacing at most $m$ points by arbitrary points of $\\R^p$.\nThe *finite-sample breakdown point* of $T$ at $x$ is\n$$\\varepsilon^*_n(T, x) = \\min\\Big\\{ \\frac{m}{n} \\;:\\; \\sup_{y \\in \\mathcal{Y}_m(x)} \\norm{T(y) - T(x)} = \\infty \\Big\\}.$$\nThe *asymptotic breakdown point* is the limit of $\\varepsilon^*_n$ as $n \\to \\infty$, when it exists.\n\n## Remarks\n\nExample: the sample median of real data has $\\varepsilon^*_n = \\floor{(n+1)/2}/n$.\nFor $n = 2k + 1$, changing $k$ points leaves an original point at the middle position, between original values; changing $k + 1$ points to a huge value $M$ makes the median $M$.\nSo the asymptotic breakdown point of the [[median|08W2CW8PKW]] is $1/2$.\n\nNon-example of robustness: the [[sample mean|0QNRBHWV35]] has $\\varepsilon^*_n = 1/n$, since replacing one point $x_1$ by $x_1 + nM$ moves the mean by $M$.\nIts asymptotic breakdown point is $0$.\n\nThe $k$-trimmed mean, which drops the $k$ smallest and $k$ largest values, has $\\varepsilon^*_n = (k+1)/n$.\nFor estimators of location that are translation equivariant, the asymptotic breakdown point is at most $1/2$.\n",
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