other M-estimator of scale, the rank of V equals 3, indicating that in this
case the IM test is sensitive to ‘more’ specification error. Explicit expres-
sions for V are derived for two robust M-estimators of scale, namely the
M-estimator based on Tukey’s biweight (TB) function and the Median Ab-
solute Deviation (MAD). Next, we examine the performance of the IM test
under four different sequences of local alternatives: a contaminated normal,
Student’s t, a skewed normal, and a tilted normal. The asymptotic distribu-
tion of the IM statistics under a sequence of local alternatives is non-central
χ2 . For the contaminated normal alternative the non-centrality parameter
is shown to be minimal, over all M-estimators, at the ML estimator. It turns
out that, when outlying observations are present, the IM test using the 25%
breakdown point TB estimator (which is a compromise between high robust-
ness and high efficiency) is much more powerful than the IM test using the
ML estimator. On the other hand, when the local alternative is Student’s
t (with degrees of freedom approaching infinity), skewed normal, or tilted
normal, the non-centrality parameter is identical for all M-estimators. We
derive closed-form expressions of the non-centrality parameter, under any
local alternative considered. We also compare, whenever relevant, the local
power of the IM tests with the local power of the score test. The latter test
is known to be optimal and so it provides a natural benchmark.
In the normal regression model the IM test is a combined test for het-
eroskedasticity, skewness and non-normal kurtosis (Hall, 1987). We use an
S-estimator (Rousseeuw and Yohai, 1984) or an MM-estimator (Yohai, 1987)
as robust estimators of regression, and an M-estimator based on Tukey’s bi-
weight function as a robust estimator of residual scale. It is well-known that
the ML estimator tends to mask outlying observations, and this danger is
more severe in the regression model than in the location-scale model. It
is therefore expected that the use of robust estimators holds more promise
in the regression case. Simulation results indeed indicate that using robust
estimators increases the power of the IM test in the case of a thick-tailed
alternative like the Cauchy distribution, or in the presence of outliers.
Note that in the presence of outliers, the misspecification test will reject
the IM equality. The aim of testing the IM equality with robust estimators is
to increase the power of the statistical test, and not to attain level-robustness
(Hampel et al., 1986). So when the null hypothesis holds, aside from some
outliers, we would like the test to detect this deviation from the specified