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AbstractMost dimension reduction methods based on nonparametric smoothing are highly sensitive to outliers and to data coming from heavy-tailed distributions. We show that the recently proposed methods by Xia et al. (2002) can be made robust in such a way that preserves all advantages of the original approach. Their extension based on the local one-step M-estimators is sufficiently robust to outliers and data from heavy tailed distributions, it is relatively easy to implement, and surprisingly, it performs as well as the original methods when applied to normally distributed data.
The Nadaraya-Watson estimator of regression is known to be highly sensitive to the presence of outliers in the sample. A possible way of robustification consists in using local L-estimates of regression. Whereas the local L-estimation is traditionally done using an empirical conditional distribution function, we propose to use instead a smoothed conditional distribution function. We show that this smoothed L-estimation approach provides computational as well as statistical finite sample improvements. The asymptotic distribution of the estimator is derived under mild β-mixing conditions.
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