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Asymptotic normality and efficiency of variance components estimators with high breakdown points

100%
Müller C.
Discussiones Mathematicae Probability and Statistics
|
2000
|
tom 20
|
nr 1
85-95
EN
For estimating the variance components of a one-way random effect model recently Uhlig (1995, 1997) and Lischer (1996) proposed non-iterative estimators with high breakdown points. These estimators base on the high breakdown point scale estimators of Rousseeuw and Croux (1992, 1993), which they called Q-estimators. In this paper the asymptotic normal distribution of the new variance components estimators is derived so that the asymptotic efficiency of these estimators can be compared with that of the maximum likelihood estimators.
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Redescending M-estimators in regression analysis, cluster analysis and image analysis

100%
Müller C.
Discussiones Mathematicae Probability and Statistics
|
2004
|
tom 24
|
nr 1
59-75
EN
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.
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