Advertisement

Uncertainty Assessment of Some Data-Adaptive M-Estimators

  • Jan Martin BrockmannEmail author
  • Boris Kargoll
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 137)

Abstract

In this paper, we review a data-adaptive class of robust estimators consisting of convex combinations of the loss functions with respect to the L 1- and Huber’s M-estimator as proposed by Dodge and :̧def :̧def Jureckova (2000). The great advantage of this approach in comparison to the traditional procedure of applying a single estimator is that the optimal weight factor, representing the data-dependent minimum-variance estimator within that class, may be estimated from the data itself. Depending on the data characteristics, one could obtain pure L 2, L 1 and Huber’s estimator, as well as any convex combination between these three. We demonstrate the computational and statistical efficiency of this approach by providing an iteratively reweighted least squares algorithm and Monte Carlo uncertainties of the weight factor.

Notes

Acknowledgements

The computations were performed on the JUMP supercomputer in Jülich. The computing time was granted by the John von Neumann Institute for Computing (project 1827)

References

  1. Barnett V, Lewis T (1994) Outliers in statistical data, 3rd edn. Wiley, ChichesterGoogle Scholar
  2. Bowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis. Oxford Statistical Science Series, vol 18. Oxford University PressGoogle Scholar
  3. Chang XW, Guo Y (2005) Huber’s M-estimation in relative GPS positioning: computational aspects. J Geodesy 79:351–362CrossRefGoogle Scholar
  4. Dodge Y, J Jureckova (2000) Adaptive regression. Springer, BerlinCrossRefGoogle Scholar
  5. Hogg RV (1974) Adaptive robust procedures: a partial review and some suggestions for future applications and theory. J Am Stat Assoc 69(348):909–923CrossRefGoogle Scholar
  6. Huber PJ (1981) Robust statistics. Wiley, New YorkGoogle Scholar
  7. Junhuan P (2005) The asymptotic variance-covariance matrix, Baarda test and the reliability of L 1-norm estimates. J Geodesy 78:668–682CrossRefGoogle Scholar
  8. Kargoll B (2005) Comparison of some robust parameter estimation techniques for outlier analysis applied to simulated GOCE mission data. In: Jekeli C et al. (eds) IAG Symposia, doi: 10.1007/3-540-26932-0_14Google Scholar
  9. Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, HeidelbergGoogle Scholar
  10. Koch KR (2007) Introduction to Bayesian statistics, 2nd edn. Springer, HeidelbergGoogle Scholar
  11. Marshall J (2002) L 1-norm pre-analysis measures for geodetic networks. J Geodesy, 76:334–344CrossRefGoogle Scholar
  12. Peracchi F (2001) Econometrics. Wiley, New YorkCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Geodesy and Geoinformation, Department of Theoretical GeodesyUniversity of BonnBonnGermany

Personalised recommendations