Robust Econometrics

one-step counterpart can be defined at sample {xi}_iⁿ₌₁ by
n /σ∙   ^θ∖  / n /■   θ⁰∖

■  '⁺ •■ XXX-(⅛^θ-)  '^x   (ɪ)■

where ■ 0 and ^0 represent initial robust estimators of location and scale
like the median and MAD, respectively. Such one-step estimators, under
certain conditions on the initial estimators, preserve the breakdown point
of the initial estimators, and at the same time, have the same first-order
asymptotic distribution as the original M -estimator (Simpson et al., 1992,
and Welsh and Ronchetti, 2002). Further development of such ideas include
an adaptive choice of parameters of function ψ in the iterative step (Gervini
and Yohai, 2002).

2.2 S-estimators

An alternative approach to M -estimators achieving high breakdown point
(HBP) was proposed by Rousseeuw and Yohai (1984). The S-estimators are
defined by minimization of a scale statistics s²(z, b) = s{η(z, b)} defined as
the M -estimate of scale,
at the model distribution F; the functions P and η are those defining M-
estimators in Section 2.1. More generally, one can define S-estimators by
means of any scale-equivariant statistics s2, that is, s{cη(z, b)} = ∣c∣s{η(z, b)}.
Under this more general definition, S-estimators include as special cases LS

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