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

Abstract

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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Acknowledgments

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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Keywords

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

Mathematics Subject Classification (2000)

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