Most asymptotic results for robust estimates rely on regularity conditions
that are difficult to verify in practice. Moreover, these results apply to
fixed distribution functions. In the robustness context the distribution of the
data remains largely unspecified and hence results that hold uniformly over a
set of possible distribution functions are of theoretical and practical
interest. Also, it is desirable to be able to determine the size of the set of
distribution functions where the uniform properties hold. In this paper we
study the problem of obtaining verifiable regularity conditions that suffice to
yield uniform consistency and uniform asymptotic normality for location robust
estimates when the scale of the errors is unknown.
We study M-location estimates calculated with an S-scale and we obtain
uniform asymptotic results over contamination neighborhoods. Moreover, we show
how to calculate the maximum size of the contamination neighborhoods where
these uniform results hold. There is a trade-off between the size of these
neighborhoods and the breakdown point of the scale estimate.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Statistics
(http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000054
