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  "body": "\n## Definition\n\n*Robust statistics* is the study of statistical procedures that remain reliable when the true distribution of the data is only close to the assumed model.\nThe basic precise notion is Hampel's qualitative robustness.\nLet $X_1, X_2, \\dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with distribution $F$, and let $T_n = T_n(X_1, \\dots, X_n)$, $n \\ge 1$, be a sequence of [[estimators|0TTNRP5PFQ]].\nWrite $\\mathcal{L}_F(T_n)$ for the law of $T_n$ when the $X_i$ have distribution $F$, and let $\\pi$ be the [[Lévy–Prokhorov metric|TASMEPWNA6]] on probability distributions.\nThe sequence $(T_n)$ is *qualitatively robust* at $F$ if for every $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that for every distribution $G$ and every $n$,\n$$\\pi(F, G) \u003c \\delta \\implies \\pi\\big(\\mathcal{L}_F(T_n), \\mathcal{L}_G(T_n)\\big) \u003c \\varepsilon.$$\n\n## Remarks\n\nQualitative robustness asks that the maps $G \\mapsto \\mathcal{L}_G(T_n)$ be continuous at $F$, uniformly in $n$.\nCloseness in $\\pi$ allows both small rounding of all data and a small fraction of arbitrary outliers.\nThe quantitative measures are the [[breakdown point|BRGGGZVQDS]] and the [[influence function|EB2QJQNE5Z]].\n\nExample: if $F$ has a unique median $m$, the sample median is qualitatively robust at $F$.\nA small fraction of outliers can shift it only to nearby order statistics, so its law changes little.\n\nNon-example: the sample mean is not qualitatively robust at any $F$.\nLet $0 \u003c \\eta \u003c \\delta$ and $G = (1 - \\eta) F + \\eta \\, \\delta_x$, with $\\delta_x$ the point mass at $x$; then $\\pi(F, G) \\le \\eta \u003c \\delta$.\nFor large $n$, with high probability about $n\\eta$ observations equal $x$, and the mean moves by about $\\eta x$, which is arbitrarily large for large $x$.\n",
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