Robust Econometrics

-^r∖ -^r∖

of an estimator θ¹ relative to another estimator θ² :

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ARE(^ ,θ2)=^as. ^var(θ.                     (7)

as. var(θ²)

For example, at the normal distribution with θ¹ and θ² being the least ab-
solute deviation (LAD) and LS estimators, ARE equals 2/π ≈ 0.64. Under
the Student cdf t5 , the ARE of the two estimators climbs up to ≈ 0.96. For
Huber’s M-estimator, we see that its limit cases are the median for c → 0
and the mean for c → ∞. At the normal distribution and for c = 1.345, we
have ARE of about 0.95. This means that this M -estimator is almost as effi-
cient as MLE, but does not lose so drastically in performance as the standard
mean under contamination because of the bounded influence function.

Whereas the influence function of M -estimators is closely related to the
choice of its objective function, the global robustness of M -estimators is in
a certain sense independent of this choice. Maronna et al. (1979) showed in
linear regression that the breakdown point of M -estimators is bounded by
1/p, where p is the number of estimated parameters. As a remedy, several
authors proposed one-step M -estimators that are defined, for example, as
the first step of the iterative Newton-Raphson procedure, used to minimize
R ρ(z,θ)dF(z), started from initial robust estimators θ⁰ of parameters and
^⁰ of scale (see Welsh and Ronchetti, 2002, for an overview). Possible initial
estimators can be those discussed in Sections 2.2 and 2.3. For example for
an M-estimator of location θ defined by a function ψ(x,θ) = ψ(x — θ), its

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