Globally Robust Inference

The robust approach to data analysis uses models that do not completely specify the distribution of the data, but rather assume that this distribution belongs to a certain neighborhood of a parametric model. Consequently, robust inference should be valid under all the distributions in these neighborhoods. Regarding robust inference, there are two important sources of uncertainty: (i) sampling variability and (ii) bias caused by outlier and other contamination of the data. The estimates of the sampling variability provided by standard asymptotic theory generally require assumptions of symmetric error distribution or alternatively known scale. None of these assumptions are met in most practical problems where robust methods are needed. One alternative approach for estimating the sampling variability is to bootstrap a robust estimate. However, the classical bootstrap has two shortcomings in robust applications. First, it is computationally very expensive (in some cases unfeasible). Second, the bootstrap quantiles are not robust. An alternative bootstrap procedure overcoming these problems is presented.The bias uncertainty is usually ignored even by robust inference procedures. The consequence of ignoring the bias can result in true probability coverage for confidence intervals much lower that the nominal ones. Correspondingly, the true significance levels of tests may be much higher than the nominal ones. We will show how the bias uncertainty can be dealt with by using maximum bias curves, obtaining confidence interval and test valid for the entire neighborhood. Applications of these ideas to location and regression models will be given. This work has been partially supported by NSERC (Canada) and Spanish Grant BFM2001-0169 (Spain).