Robust Econometrics

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 ^{+ εδ}x} - 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 {x_i}n=₁, we obtain

г; T, Fn ) = lim
ε→0

= lim

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)

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