| Summary: | We study the problem of performing statistical inference based on robust estimates
when the distribution of the data is only assumed to belong to a contamination
neighbourhood of a known central distribution. We start by determining the asymptotic
properties of some robust estimates when the data are not generated by the
central distribution of the contamination neighbourhood. Under certain regularity
conditions the considered estimates are consistent and asymptotically normal. For
the location model and with additional regularity conditions we show that the convergence
is uniform on the contamination neighbourhood. We determine that a class of
robust estimates satisfies these requirements for certain proportions of contamination,
and that there is a trade-off between the robustness of the estimates and the extent
to which the uniformity of their asymptotic properties holds. When the distribution
of the data is not the central distribution of the neighbourhood the asymptotic variance
of these estimates is involved and difficult to estimate. This problem affects the
performance of inference methods based on the empirical estimates of the asymptotic
variance. We present a new re-sampling method based on Efron's bootstrap (Efron,
1979) to estimate the sampling distribution of MM-location and regression estimates.
This method overcomes the main drawbacks of the use of bootstrap with robust estimates
on large and potentially contaminated data sets. We show that our proposal is
computationally simple and that it provides stable estimates when the data contain
outliers. This new method extends naturally to the linear regression model. === Science, Faculty of === Statistics, Department of === Graduate
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