and for which ρc(0) = 0. This estimator is consistent for σ and has influence
function (Huber, 1981)
IF(u; σ∙,Φ) = Pc(^> - bc .
E φ [ pc ( u ) u ]
For ^ to be robust, ρc has to be bounded and / has to be a robust estimator
of location (e.g. the median). The ML estimator of σ, for example, is an
M-estimator of scale for which ρc(u) = u2 (unbounded), bc = 1, and / is the
sample average, and hence is non-robust.
of scale
defined by Tukey’s biweight
Consider the robust M-estimator
(TB) function,
{u 2
2
C2
6 ,
u 4
2c2
+ u6
+ 6c4,
if |u| ≤c;
if |u| >c.
The choice of c is governed by the desired breakdown point of ^. The details
of how to compute T , for any choice of c, are given in Appendix A. Table 1
gives the numerical results for 10%, 25% and 50% breakdown points that
are needed to compute T using the TB estimator.
Another, simpler, robust M-estimator of scale is the (standardized) Me-
dian Absolute Deviation (MAD),
σ = Ф-1(3/4) medi{lYi - medjYj|}’
for which ρc(u) = I(|u| ≥ c) with c = Φ-1(3/4) = 0.6745 and bc = 1 /2. The
breakdown point of the MAD is 50%. Table 1 gives the constants, derived
in Appendix A, that are needed to compute T using the MAD.
4 Local asymptotic power
Let Fn be a sequence of local alternatives tending to the null distribution,
i.e. Fn → Fθ. Under a sequence of local alternatives,
Hn : Y ~ Fn,
the IM indicator M is, given some regularity conditions, asymptotically
normally distributed,
n(M ( M - mn ) 4 N(0 ,v ),