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.
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
Notes
- 1.
I( ⋅) denotes the indicator function, which returns 1 if the statement in the argument is true and 0 otherwise.
References
Barnett V, Lewis T (1994) Outliers in statistical data, 3rd edn. Wiley, Chichester
Bowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis. Oxford Statistical Science Series, vol 18. Oxford University Press
Chang XW, Guo Y (2005) Huber’s M-estimation in relative GPS positioning: computational aspects. J Geodesy 79:351–362
Dodge Y, J Jureckova (2000) Adaptive regression. Springer, Berlin
Hogg RV (1974) Adaptive robust procedures: a partial review and some suggestions for future applications and theory. J Am Stat Assoc 69(348):909–923
Huber PJ (1981) Robust statistics. Wiley, New York
Junhuan P (2005) The asymptotic variance-covariance matrix, Baarda test and the reliability of L 1-norm estimates. J Geodesy 78:668–682
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_14
Koch KR (1999) Parameter estimation and hypothesis testing in linear models. Springer, Heidelberg
Koch KR (2007) Introduction to Bayesian statistics, 2nd edn. Springer, Heidelberg
Marshall J (2002) L 1-norm pre-analysis measures for geodetic networks. J Geodesy, 76:334–344
Peracchi F (2001) Econometrics. Wiley, New York
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Brockmann, J.M., Kargoll, B. (2012). Uncertainty Assessment of Some Data-Adaptive M-Estimators. In: Sneeuw, N., Novák, P., Crespi, M., Sansò, F. (eds) VII Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, vol 137. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22078-4_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-22078-4_13
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22077-7
Online ISBN: 978-3-642-22078-4
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)