The influence of infinitesimal contamination on an estimator is charac-
terized by the influence function, which measures the relative change in es-
timates caused by an infinitesimally small amount ε of contamination at x
(Hampel et al., 1986). More formally,
IF(x; T, F) = lim T{(1 - ε)F + εδx} - T(F). (1)
ε→0 ε
For each point г, the influence function reveals the rate at which the esti-
mator T changes if a wrong observation appears at г. In the case of sample
mean x = T(Fn) for {xi}n=1, we obtain
(1 - ε) udFn(u) + ε udδx (u) - udFn (u) /ε
г; T, Fn ) = lim
ε→0
= lim
- udFn(u) + udδx (u)
ε→0
The influence function allows us to define various desirable properties
of an estimation method. First, the largest influence of contamination on
estimates can be formalized by the gross-error sensitivity,
γ(T,F) = supIF(x;T,F), (2)
x∈R
which under robustness considerations be finite and small. Even though such
a measure can depend on F in general, the qualitative results (e.g., γ(T, F)
being bounded) are typically independent of F. Second, the sensitivity to
small changes in data, for example moving an observation from x to y ∈ R,
can be measured by the local-shift sensitivity
λT Fy-F ( χ ; T,F ) - 1F ( y ; T-F ) k rn
λ ( T’F ) = ----------iF-yii----------• (3)