For the computation of the IM test statistic with robust estimators (in par-
ticular, for computing B), we refer to Section 3.2, since all robust scale
estimators considered here are M-estimators based on Tukey’s biweight func-
tion. We consider two robust estimators of β. The first one is the S-estimator
(Rousseeuw and Yohai, 1984),
A ∙ ! ∏∖
β1 = arg min s(β),
β
where s(β) is a robust M-estimator of scale, i.e. it solves
1 ʌ YY- - X,iβ∖h
n - ρ 1∖ s ( β ) =b bc1 '
where ρc and bc are as in Section 3.2. The second one is the MM-estimator
(Yohai, 1987), which solves
where ^ is an initial residual scale estimator based on a very robust S-
estimator, i.e. σ = s(∣31 ). The constant c2 is chosen large enough to obtain
an increase in efficiency upon the S-estimator. By selecting c1 and c2 appro-
priately, this MM-estimator combines the high breakdown property (25%)
with a higher statistical efficiency (95% at Gaussian models) than the S-
estimator.
A
β2 = arg min
β
A /Y
2 ( -
i=1
- X- β
ʌ
^
6 Monte Carlo results
6.1 The normal model
In Section 4 we studied the local behaviour of the IM test under various
alternatives. Here the finite sample power (against fixed alternatives) is
investigated by means of Monte Carlo experiments.
We look at three alternative hypotheses: a normal distribution contam-
inated with outliers, the Cauchy distribution, and the χ2 distribution. To
estimate the location robustly, we use the median, which has a breakdown
point of 50%. As robust scale estimators we use the MAD and the TB M-
estimator with 25% breakdown point. We carry out 10000 simulations. To
20