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Procedures for local-constant smoothing are investigated in a broad variety of data situations with outliers and jumps. Moving window and nearest neighbour versions of mean and median smoothers are considered, as well as double window and linear hybrid smoothers. For the choice of the window width or the number of neighbours the different estimators are combined with each of several cross-validation criteria like least squares, least absolute deviations, and median-crossvalidation. It is identified, which method works best in which data scenarios. Although there is not a single overall best robust smoothing procedure, a robust cross-validation criterion, called least trimmed squares-cross-validation, gives good results for most smoothing methods and data situations, with cross-validation based on least absolute deviations being a strong competitor, particularly if there are jumps, but little problems with outliers in the data.

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