on one or more constants a, b, c ∈ R. If an estimator T is to be invariant
to the scale of data, one can apply the estimator to rescaled data, that is,
to minimize ʃ ρ{(z — θ)/s}dFn(z) or to solve ʃ ψ{(z — θ)/s}dFn(z) = 0 for
a scale estimate s like the median absolute deviation (MAD). Alternatively,
one may also estimate parameters θ and scale s simultaneously by considering
P(z, {θ,s}) = P{(z - θ)/s} or
ψ(z, {θ, s}) = {ψl(z, {θ, s}), ψs(z, {θ, s})}.
Let us now turn to the question how the choice of functions ρ and ψ deter-
mines the robust properties of M -estimators. First, the influence function of
an M -estimator can generally depend on several quantities such as its asymp-
totic variance or the position of explanatory variables in the regression case,
but the influence function is always proportional to function ψ(z, b). Thus,
the finite gross-error sensitivity, γ(T, F) < ∞, requires bounded ψ(t) (which
is not the case of LS). Similarly, the finite rejection point, ρ(T, F) < ∞, leads
to ψ(t) being zero for all sufficiently large t (the M -estimators defined by such
a ψ-function are called redescending). Hampel et al. (1986) shows how, for
a given bound on γ(T, F), one can determine the most efficient choice of ψ
function (e.g., the skipped median, ψ(t) = sign(t)I (|t| < K), K > 0, in the
location case).
More formally, the optimality of M -estimators in the context of qualita-
tive robustness can be studied by the asymptotic relative efficiency (ARE)
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