Influence function

Remarks

It is the one-sided Gateaux derivative of $T$ at $F$ in the direction $\delta_x - F$. The 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. Under 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.

Example: the mean $T(F) = \int t \, dF(t)$, on distributions with finite mean. Then $T((1-\varepsilon)F + \varepsilon\delta_x) = (1-\varepsilon)\mu + \varepsilon x$ with $\mu = T(F)$, so $\mathrm{IF}(x; T, F) = x - \mu$, which is unbounded. For 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.

Non-example: the mean is not defined at a distribution $F$ without a finite expectation, such as the Cauchy distribution. Then $F$ lies outside $\mathcal{D}$, and the mean has no influence function at $F$.