On robust cross-validation for nonparametric smoothing

An essential problem in nonparametric smoothing of noisy data is a proper choice of the bandwidth or window width, which depends on a smoothing parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation>. One way to choose <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation> based on the data is leave-one-out-cross-validation. The selection of the cross-validation criterion is similarly important as the choice of the smoother. Especially, when outliers are present, robust cross-validation criteria are needed. So far little is known about the behaviour of robust cross-validated smoothers in the presence of discontinuities in the regression function. We combine different smoothing procedures based on local constant fits with each of several cross-validation criteria. These combinations are compared in a simulation study under a broad variety of data situations with outliers and abrupt jumps. There is not a single overall best cross-validation criterion, but we find Boente-cross-validation to perform well in case of large percentages of outliers and the Tukey-criterion in case of data situations with jumps, even if the data are contaminated with outliers. Copyright Springer-Verlag Berlin Heidelberg 2013