Consistency of completely outlier-adjusted simultaneous redescending M-estimators of location and scale |
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Authors: | Martin Bachmaier |
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Institution: | 1.Fachgebiet Technik im Pflanzenbau,Technische Universit?t München,Freising,Germany |
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Abstract: | This paper gives conditions for the consistency of simultaneous redescending M-estimators for location
and scale. The consistency postulates the uniqueness of the parameters μ and σ, which are defined
analogously to the estimations by using the population distribution function instead of the empirical
one. The uniqueness of these parameters is no matter of course, because redescending ψ- and χ-functions,
which define the parameters, cannot be chosen in a way that the parameters can be considered as
the result of a common minimizing problem where the sum of ρ-functions of standardized residuals
is to be minimized. The parameters arise from two minimizing problems where the result of one problem
is a parameter of the other one. This can give different solutions. Proceeding from a symmetrical
unimodal distribution and the usual symmetry assumptions for ψ and χ leads, in most but not
in all cases, to the uniqueness of the parameters. Under this and some other assumptions, we can also
prove the consistency of the according M-estimators, although these estimators are usually not unique
even when the parameters are. The present article also serves as a basis for a forthcoming
paper, which is concerned with a completely outlier-adjusted confidence interval for μ. So
we introduce a ñ where data points far away from the bulk of the data are not counted at
all. |
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Keywords: | Consistent Robust Redescending M-estimate M-estimator Unique Location parameter Scale parameter Outlier Gross error |
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