Design of kernel M-smoothers for spatial data

Robust nonparametric smoothers have been proved effective to preserve edges in image denoising. As an extension, they should be capable to estimate multivariate surfaces containing discontinuities on the basis of a random spatial sampling. A crucial problem is the design of their coefficients, in particular those of the kernels which concern robustness. In this paper it is shown that bandwidths which regard smoothness can consistently be estimated, whereas those which concern robustness cannot be estimated with plug-in and cross-validation criteria. Heuristic and graphical methods are proposed for their selection and their efficacy is proved in simulation experiments.