Summary
Since recent interests are to consider not just one solution but all possible solutions to the redescending M estimating equations to identify possible multiple structure in a data set (Morgenthaler, 1990; Meer and Tyler, 1998), the focus in this paper is the re-descending M estimators for regression. We use multiple local minima for finding particular structures, such as lines, in the data set by associating them with the local minima in a re-descending M estimation problem for regression. A simple qualitative measure is constructed to assess the importance of each local minimum point, and hence importance of each fit to the data. This testing procedure allows us to distinguish between good and bad fits to the data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
O. Arslan. Convergence behavior of the iteratively reweighting algorithm to compute the multivariate M estimates for location and scatter. Submitted, 1999.
O. Arslan, O. Edlund, and H. Ekblom. Algorithms to compute CM- and S-estimates for regression. Technical report, Dept. of Mathematics, Lule A University of Technology, Sweden, 2001.
M.J. Black and A. Rangarajan. On the unification of line processes, outlier rejection, and robust statistics with application in early vision. Int. J. of Computer Vision, 19(1):57–91, 1996.
F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel. Robust statistics: The approach based on influence functions. Wiley, New York, 1986.
J.T. Kent and D.E. Tyler. Constrained M-estimation for multivariate location and scatter. Ann. Statist., 24(3):1346–1370, 1996.
M. Markatou, A. Basu, and B.G. Lindsay. Weighted likelihood estimating equations with a bootstrap root search. J. Am. Statist. Assoc., 93:740–750, 1998.
R.A. Maronna and V.J. Yohai. Asymptotic behavior of generalized M-estimates for regression and scale with random carriers. Z Wahrscheinlichkeitstheorie verw. Gebiete,58:7–20, 1981.
P. Meer and D.E. Tyler. Smoothing the gap between statistics and image understanding. Comments on the paper “Edge-preserving smothers for image processing” by C.K. Chu, I.K. Glad, F. Godtlibsen, and J.S. Marron. J. Am. Statist. Assoc.,93(442):548–553, 1998.
B. Mendes and D.E. Tyler. Constrained M estimation for regression. In Robust Statistics,Data Analysis and Computer Intensive Methods,pages 299–320. Springer, New York, 1995.
B. Mendes and D.E. Tyler. Bias robustness of the CM-estimates of location and scale. Brazilian J. of Probability and Statistics, 12:41–53, 1998.
S. Morgenthaler. Fitting redescending M-estimators in regression. In K.D. Lawrence and J.L. Arthur, editors, Robust Regression: Analysis and Applications, chapter 5, pages 105–128. Dekker, New York, 1990.
G. Roth and M.D. Levine. Extracting geometric primitive. CVGIP: Image Understanding, 58(1):1–22, 1993.
P.J. Rousseeuw and V.J. Yohai. Robust regression by means of S-estimators. In J. Frank, W. Hardie, and R.D. Martin, editors, Robust and Nonlinear Time Series Analysis. Lecture Notes in Statistics,pages 256–272. Springer, New York, 1984.
C.G. Small, J. Wang, and Z. Yang. Eliminating multiple root problems in estimation. Statistical Science, 15:313–341, 2000.
C.V. Stewart. Bias in robust estimation caused by discontinuous and multiple structures. IEEE Trans. on PAMI,19:818–833, 1997.
K.S. Tatsuoka. M, CM and S-estimates: Theory and computation. PhD thesis, Rutgers University, Dept. of Statist., New Jersey, USA, 1996.
D.E. Tyler. Some issues in the robust estimation of multivariate location and scatter. In W. Stahel and S. Weisberg, editors, Directions in Robust Statistics and Diagnostics 2, pages 327–336. Springer, New York, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arslan, O. (2003). A Simple Test to Identify Good Solutions of Redescending M Estimating Equations for Regression. In: Dutter, R., Filzmoser, P., Gather, U., Rousseeuw, P.J. (eds) Developments in Robust Statistics. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-57338-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-57338-5_4
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-642-63241-9
Online ISBN: 978-3-642-57338-5
eBook Packages: Springer Book Archive