Local-shift sensitivity (${\lambda }^{\ast }(T,F)$)

The local-shift sensitivity of a scalar statistical functional isIt measures the sensitivity of the first-order contamination effect to moving the contaminating observation. If the influence function is continuously differentiable with bounded derivative, the mean value theorem gives ${\lambda }^{\ast }\le {\sup }_x|{\mathrm{IF}}^{\prime }(x)|$, and taking $y\to x$ proves equality. A jump in the influence function makes ${\lambda }^{\ast }$ infinite, even when gross-error sensitivity is finite.