Discussiones Mathematicae Probability and Statistics 24(1) (2004) 59-75


Christine H. Müller

Carl von Ossietzky University Oldenburg
Institute for Mathematics
Postfach 2503, D-26111 Oldenburg, Germany


We give a review on the properties and applications of M-estimators with redescending score function. For regression analysis, some of these redescending M-estimators can attain the maximum breakdown point which is possible in this setup. Moreover, some of them are the solutions of the problem of maximizing the efficiency under bounded influence function when the regression coefficient and the scale parameter are estimated simultaneously. Hence redescending M-estimators satisfy several outlier robustness properties. However, there is a problem in calculating the redescending M-estimators in regression. While in the location-scale case, for example, the Cauchy estimator has only one local extremum this is not the case in regression. In regression there are several local minima reflecting several substructures in the data. This is the reason that the redescending M-estimators can be used to detect substructures in data, i.e. they can be used in cluster analysis. If the starting point of the iteration to calculate the estimator is coming from the substructure then the closest minimum corresponds to this substructure. This property can be used to construct an edge and corner preserving smoother for noisy images so that there are applications in image analysis as well.

Keywords: redescending M-estimator, regression, breakdown point, optimality, cluster analysis, image analysis, kernel estimator.

2000 Mathematics Subject Classification: 62J05, 62G35, 62H30, 62G07, 62M40, 62P99.


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Received 4 October 2003