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Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness

Content Provider	Springer Nature Link
Author	Xu, Peiliang
Copyright Year	2005
Abstract	The findings of this paper are summarized as follows: (1) We propose a sign-constrained robust estimation method, which can tolerate 50% of data contamination and meanwhile achieve high, least-squares-comparable efficiency. Since the objective function is identical with least squares, the method may also be called sign-constrained robust least squares. An iterative version of the method has been implemented and shown to be capable of resisting against more than 50% of contamination. As a by-product, a robust estimate of scale parameter can also be obtained. Unlike the least median of squares method and repeated medians, which use a least possible number of data to derive the solution, the sign-constrained robust least squares method attempts to employ a maximum possible number of good data to derive the robust solution, and thus will not be affected by partial near multi-collinearity among part of the data or if some of the data are clustered together; (2) although M-estimates have been reported to have a breakdown point of 1/(t+1), we have shown that the weights of observations can readily deteriorate such results and bring the breakdown point of M-estimates of Huber’s type to zero. The same zero breakdown point of the L 1-norm method is also derived, again due to the weights of observations; (3) by assuming a prior distribution for the signs of outliers, we have developed the concept of subjective breakdown point, which may be thought of as an extension of stochastic breakdown by Donoho and Huber but can be important in explaining real-life problems in Earth Sciences and image reconstruction; and finally, (4) We have shown that the least median of squares method can still break down with a single outlier, even if no highly concentrated good data nor highly concentrated outliers exist.
Starting Page	146
Ending Page	159
Page Count	14
File Format	PDF
ISSN	09497714
Journal	Journal of Geodesy
Volume Number	79
Issue Number	1-3
e-ISSN	14321394
Language	English
Publisher	Springer-Verlag
Publisher Date	2005-06-02
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Adaptive trimmed mean Breakdown point L 1-norm Least median of squares Outlier Repeated median Robust estimation Robustness Sign-constrained robust least squares Geophysics/Geodesy Math. Applications in Geosciences
Content Type	Text
Resource Type	Article
Subject	Geophysics Geochemistry and Petrology Computers in Earth Sciences

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Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness

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