Tukey’s M-estimator of the Poisson parameter with a special focus on small means


We treat robust M-estimators for independent and identically distributed Poisson data. We introduce modified Tukey M-estimators with bias correction and compare them to M-estimators based on the Huber function as well as to weighted likelihood and other estimators by simulation in case of clean data and data with outliers. In particular, we investigate the problem of combining robustness and high efficiencies at small Poisson means caused by the strong asymmetry of such Poisson distributions and propose a further estimator based on adaptive trimming. The advantages of the constructed estimators are illustrated by an application to smoothing count data with a time varying mean and level shifts.

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  1. Agostinelli C (2013) Wle. Version 0.9-9. Available from http://cran.r-project.org/

  2. Bernholt T, Fried R, Gather U, Wegener I (2006) Modified repeated median filters. Stat Comput 16:177–192

    MathSciNet  Article  MATH  Google Scholar 

  3. Cadigan NG, Chen J (2001) Properties of robust M-estimators for Poisson and negative binomial data. J Stat Comput Simul 70:273–288

  4. Cantoni E, Ronchetti E (2001) Robust inference for generalized linear models. J Am Stat Assoc 96:1022–1030

    MathSciNet  Article  MATH  Google Scholar 

  5. Chu CK, Glad IK, Godtliebsen F, Marron JS (1998) Edge-preserving smoothers for image processing. J Am Stat Assoc 93:526–541

  6. Hampel F (1974) The influence curve and its role in robust estimation. J Am Stat Assoc 69:383–393

    MathSciNet  Article  MATH  Google Scholar 

  7. Huber P (1981) Robust statistic of location parameter. Wiley, New York

    Google Scholar 

  8. Kohl M, Ruckdeschel P (2010) R package distrMod: object-oriented implementation of probability models. J Stat Softw 35:1–27

    Article  Google Scholar 

  9. Kohl M, Ruckdeschel P (2013) ROptEst. Version 0.9. Available from http://cran.r-project.org/

  10. Lee Y, Kassam S (1985) Generalized median filtering and related nonlinear filtering techniques. IEEE Trans Acoust Speech Signal Process 33:672–683

    Article  Google Scholar 

  11. Markatou M, Basu A, Lindsay B (1997) Weighted likelihood estimating equations: the discrete case with applications to logistic regression. J Stat Plan Inference 57:215–232

    MathSciNet  Article  MATH  Google Scholar 

  12. Marazzi A, Yohai V (2010) Optimal robust estimates using the Hellinger distance. Adv Data Anal Classif 4:169–179

    MathSciNet  Article  MATH  Google Scholar 

  13. Maronna RA, Martin RD, Yohai VJ (2006) Robust Statistics. Wiley, New York

    Google Scholar 

  14. R version 3.0.2 (2013) Copyright (C) 2013 The R Foundation for Statistical Computing

Download references


The FCS data are kindly provided by the Department of Systemic Cell Biology of the Max Planck Institute of Molecular Physiology in Dortmund, Germany. The authors are very grateful for the stimulating comments of two reviewers and the editor which were very helpful for improving the paper.

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Correspondence to Roland Fried.

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Elsaied, H., Fried, R. Tukey’s M-estimator of the Poisson parameter with a special focus on small means. Stat Methods Appl 25, 191–209 (2016). https://doi.org/10.1007/s10260-015-0295-x

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  • Count data
  • Poisson model
  • Huber M-estimator
  • Tukey M-estimator
  • Outliers
  • Robustness

Mathematics Subject Classification (2000)

  • 62-07
  • 62F35
  • 62G05
  • 62G35