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  "body": "\n## Definition\n\nLet $\\mathcal{D}$ be a convex set of [[probability distributions|0VGMEBVW1M]] on $\\R$ that contains every [[point mass|099SXAG014]] $\\delta_x$, and let $T \\colon \\mathcal{D} \\to \\R$ be a functional.\nLet $F \\in \\mathcal{D}$ and $x \\in \\R$.\nThe *influence function* of $T$ at $F$ is\n$$\\mathrm{IF}(x; T, F) = \\lim_{\\varepsilon \\downarrow 0} \\frac{T\\big((1 - \\varepsilon) F + \\varepsilon \\delta_x\\big) - T(F)}{\\varepsilon},$$\ndefined at those $x$ where the limit exists.\n\n## Remarks\n\nIt is the one-sided [[Gateaux derivative|PDXR52YGPD]] of $T$ at $F$ in the direction $\\delta_x - F$.\nThe gross-error sensitivity is $\\gamma^*(T, F) = \\sup_x \\abs{\\mathrm{IF}(x; T, F)}$, and $T$ is called B-robust at $F$ when $\\gamma^*$ is finite.\nUnder regularity conditions, $\\sqrt{n}\\big(T(F_n) - T(F)\\big)$ is asymptotically normal with mean $0$ and variance $E_F\\big[\\mathrm{IF}(X; T, F)^2\\big]$, where $F_n$ is the empirical distribution.\n\nExample: the mean $T(F) = \\int t \\, dF(t)$, on distributions with finite mean.\nThen $T((1-\\varepsilon)F + \\varepsilon\\delta_x) = (1-\\varepsilon)\\mu + \\varepsilon x$ with $\\mu = T(F)$, so $\\mathrm{IF}(x; T, F) = x - \\mu$, which is unbounded.\nFor the median at a distribution $F$ with median $m$ and a density $f$ that is continuous and positive at $m$, $\\mathrm{IF}(x; T, F) = \\sgn(x - m) / (2 f(m))$ for $x \\ne m$, which is bounded.\n\nNon-example: the mean is not defined at a distribution $F$ without a finite [[expectation|0DCT0QZVQ1]], such as the Cauchy distribution.\nThen $F$ lies outside $\\mathcal{D}$, and the mean has no influence function at $F$.\n",
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