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,
У ρ ι[ η ( z,θ )/s{η ( z,θ )} ] dFn ( z )
=K=
ρ1(t)dF (t).
Next, the τ -estimate of scale is defined by
(z,θ)=s2
{η(z, θ)}
ρ2[η(z, θ)/s{η(z, θ)}]dFn(z)
and the corresponding τ -estimator of parameters θ is then defined by mini-
mizing the τ -estimate of scale, τ2(z, θ).
15