Redescending M-estimators and Deterministic Annealing, with Applications to Robust Regression and Tail Index Estimation

Authors

  • Rudolf Frühwirth Austrian Academy of Sciences, Vienna, Austria
  • Wolfgang Waltenberger Austrian Academy of Sciences, Vienna, Austria

DOI:

https://doi.org/10.17713/ajs.v37i3&4.310

Abstract

A new type of redescending M-estimators is constructed, based on data augmentation with an unspecified outlier model. Necessary and sufficient conditions for the convergence of the resulting estimators to the Hubertype skipped mean are derived. By introducing a temperature parameter the concept of deterministic annealing can be applied, making the estimator insensitive to the starting point of the iteration. The properties of the annealing
M-estimator as a function of the temperature are explored. Finally, two applications
are presented. The first one is the robust estimation of interaction vertices in experimental particle physics, including outlier detection. The second one is the estimation of the tail index of a distribution from a sample using robust regression diagnostics.

References

Atkinson, A., and Riani, M. (2000). Robust Diagnostic Regression Analysis. New York: Springer.

Beirlant, J., Vynckier, P., and Teugels, J. L. (1996). Tail index estimation, Pareto quantile plots, and regression diagnostics. Journal of the American Statistical Asssociation, 91, 1659.

Bickel, D. R., and Frühwirth, R. (2006). On a fast, robust estimator of the mode: Comparisons to other robust estimators with applications.

CMS collaboration. (1994). CMS Technical Proposal (Tech. Rep.). (Technical Report CERN/LHCC 94-38, CERN, Geneva)

CMS Collaboration. (2007). CMS Detector Information. (url:

http://cmsinfo.cern.ch/outreach/CMSdetectorInfo/CMSdetectorInfo.html)

Coreless, R. M., Gonnet, G. H., Hare, D. E. G., Jerrey, D. J., and Knuth, D. E. (1996). On the Lambert W Function.

Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1.

Frühwirth, R., and Strandlie, A. (1999). Track fitting with ambiguities and noise: a study of elastic tracking and nonlinear filters. Computer Physics Communications, 120, 197.

Garlipp, T., andMüller, C. (2005). Regression clustering with redescending M-estimators. In D. Baier and K. Wernecke (Eds.), Innovations in Classification, Data Science, and Information Systems. Berlin, Heidelberg, New York: Springer.

Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. New York: John Wiley & Sons.

Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3, 1163.

Huber, P. J. (2004). Robust Statistics: Theory and Methods. New York: John Wiley & Sons.

Li, S. Z. (1996). Robustizing robust M-estimation using deterministic annealing. Pattern recognition, 29, 159.

Müller, C. (2004). Redescending M-estimators in regression analysis, cluster analysis and image analysis. Discussiones Mathematicae — Probability and Statistics, 24, 59.

Rose, K. (1998). Deterministic annealing for clustering, compression, classification, regression, and related optimization problems.

Rousseeuw, P. J., and Leroy, A. M. (1987). Robust Regression and Outlier Detection.

Seneta, E. (1976). Regularly Varying Functions. Berlin, Heidelberg, New York: Springer.

Waltenberger, W., Frühwirth, R., and Vanlaer, P. (2007). Adaptive vertex fitting. Journal of Physics G: Nuclear and Particle Physics, 34, 343.

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Published

2016-04-03

How to Cite

Frühwirth, R., & Waltenberger, W. (2016). Redescending M-estimators and Deterministic Annealing, with Applications to Robust Regression and Tail Index Estimation. Austrian Journal of Statistics, 37(3&4), 301–317. https://doi.org/10.17713/ajs.v37i3&4.310

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