Robust Econometrics

methods. Nevertheless, if an S-estimator is not applied directly to sam-
ple observations, but rather to the set of all pairwise differences of sample
observations, the resulting generalized S-estimator exhibits higher relative
efficiency for Gaussian data, while preserving its robust properties (Croux et
al., 1994; Stromberg et al., 2000).

2.3 τ -estimators

The S-estimators improve upon M -estimators in terms of their breakdown-
point properties, but at the cost of low Gaussian efficiency. Although one-
step M -estimators based on an initial S-estimate can remedy this deficiency
to a large extent, their exact breakdown properties are not known. One of
alternative approaches, proposed by Yohai and Zamar (1988), extend the
principle of S-estimation in the following way. Assuming that ρ1 and ρ2
are non-negative, even, and continuous functions, the M -estimate s² (z, θ) =
s²{η(z, θ)} of scale can be defined as in the case of S-estimation,

Next, the τ -estimate of scale is defined by

and the corresponding τ -estimator of parameters θ is then defined by mini-
mizing the τ -estimate of scale, τ²(z, θ).

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