x = T(Fn) has the following properties. First, the influence function (1)
|
IF (x; T, F) = = |
T{(1-ε)F+εδx}-T(F) ε→0 ε {(1 - ε) R udF (u) + εx} - R udF (u) ε→0 ε lεi→m0 ε-1{-ε udF (u) + εx} x-Z udF(u)=x-T(F). |
Hence, the gross-error sensitivity (2) γ(T, F) = ∞, the local-shift sensitivity
(3) λ(T, F) = 0, and the rejection point (4) ρ(T, F) = ∞. Second, the
maximum-bias (5) is infinite for any ε > 0 since
suRp kT {(1 - ε)F + εδx} - T(F)k = suRp k - εT(F) + εxk = ∞.
Consequently, the breakdown point (6) of the sample mean x = T(Fn) is
zero, ■(T) = 0.
Thus, none of robustness measures characterizing the change of T under
contamination of data (even infinitesimally small) is finite. This behavior,
typical for LS-based methods, motivated alternative estimators that have the
desirable robust properties. In this section, the M -estimators, S-estimators,
and τ -estimators are discussed as well as some extensions and combination
of these approaches. Even though there is a much wider range of robust esti-
mation principles, we focus on those already studied and adopted in various
areas of econometrics.