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=== Desirable properties ===

Properties of an influence function that bestow it with desirable performance are:
# Finite rejection point <math>\rho^*</math>,
# Small gross-error sensitivity <math>\gamma^*</math>,
# Small local-shift sensitivity <math>\lambda^*</math>.

==== Rejection point ====
: <math>\rho^* := \inf_{r>0}\{r:IF(x;T;F)=0, |x|>r\}</math>

==== Gross-error sensitivity ====
: <math>\gamma^*(T;F) := \sup_{x\in\mathcal{X}}|IF(x; T ; F)|</math>

==== Local-shift sensitivity ====
: <math>\lambda^*(T;F) := \sup_{(x,y)\in\mathcal{X}^2\atop x\neq y}\left\|\frac{IF(y ; T; F) - IF(x; T ; F)}{y-x}\right\|</math>

This value, which looks a lot like a [[Lipschitz constant]], represents the effect of shifting an observation slightly from <math>x</math> to a neighbouring point <math>y</math>, i.e., add an observation at <math>y</math> and remove one at <math>x</math>.