M-estimation using unbiased median variance estimate

This paper first proves that the traditional median variance estimate is biased when the sample number is small and then proposes an unbiased median variance estimate to calibrate for the bias of the variance estimate. The scaled median variance estimate is firstly derived, and the unbiased median variance estimate is formed with independent residuals in an adjustment model no matter whether the measurements are contaminated by outliers or not. Using the unbiased median variance estimate, the M-estimate is constructed to mitigate for the biases caused by the variance estimate. The IGGIII reduction factor is used to verify the proposed algorithms by a levelling network example. Numerical analysis confirms that the proposed median variance estimate can achieve better unbiasedness for contaminated measurement set, but the dispersion of our estimate is unfortunately larger than that for the least-squares estimate.

[1]  J. Tukey,et al.  The Fitting of Power Series, Meaning Polynomials, Illustrated on Band-Spectroscopic Data , 1974 .

[2]  S. Hekimoglu.  Robustifying conventional outlier detection procedures , 1999 .

[3]  Edward E. Cureton,et al.  Unbiased Estimation of the Standard Deviation , 1968 .

[4]  P. Teunissen.  Testing Theory: an introduction , 2009 .

[5]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[6]  Peter Teunissen,et al.  Minimal detectable biases of GPS data , 1998 .

[7]  Peiliang Xu.  Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness , 2005 .

[8]  Dennis K. J. Lin,et al.  Robust recursive estimation for correlated observations , 1995 .

[9]  W. Baarda,et al.  Statistical concepts in geodesy. , 1967 .

[10]  Liangjun Su.  Advanced Mathematical Statistics , 2007 .

[11]  P. Rousseeuw.  Least Median of Squares Regression , 1984 .

[12]  Tianhe Xu,et al.  Robust estimator for correlated observations based on bifactor equivalent weights , 2002 .

[13]  Karl-Rudolf Koch,et al.  Parameter estimation and hypothesis testing in linear models , 1988 .

[14]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[15]  Ling Yang,et al.  Outlier separability analysis with a multiple alternative hypotheses test , 2013, Journal of Geodesy.

[16]  K. Koch.  Deviations from the null hypothesis to be detected by statistical tests , 1981 .

[17]  P. J. Huber.  Robust Estimation of a Location Parameter , 1964 .

[18]  W. Baarda,et al.  A testing procedure for use in geodetic networks. , 1968 .

[19]  Rüdiger Lehmann,et al.  Improved critical values for extreme normalized and studentized residuals in Gauss–Markov models , 2012, Journal of Geodesy.

[20]  K. Koch.  Robust estimation by expectation maximization algorithm , 2013, Journal of Geodesy.

[21]  Karl-Rudolf Koch,et al.  Minimal detectable outliers as measures of reliability , 2015, Journal of Geodesy.

[22]  Michael Spann,et al.  Robust Optical Flow Computation Based on Least-Median-of-Squares Regression , 1999, International Journal of Computer Vision.

[23]  P. J. Huber.  Finite Sample Breakdown of $M$- and $P$-Estimators , 1984 .

[24]  Peiliang Xu.  Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness , 2005 .

[25]  Peter J. Rousseeuw,et al.  ROBUST REGRESSION BY MEANS OF S-ESTIMATORS , 1984 .

[26]  S. Ross.  A First Course in Probability , 1977 .

[27]  Burkhard Schaffrin,et al.  Reliability Measures for Correlated Observations , 1997 .

[28]  Serif Hekimoglu,et al.  Finite Sample Breakdown Points of Outlier Detection Procedures , 1997 .

[29]  Byron D. Tapley,et al.  Robust estimation of systematic errors of satellite laser range , 1999 .

[30]  Werner A. Stahel,et al.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[31]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[32]  Peter J. Rousseeuw,et al.  Robust Regression and Outlier Detection , 2005, Wiley Series in Probability and Statistics.

[33]  Michael Lösler,et al.  Multiple Outlier Detection: Hypothesis Tests versus Model Selection by Information Criteria , 2016 .

[34]  W. Prószyñski.  On outlier-hiding effects in specific Gauss–Markov models: geodetic examples , 2000 .

[35]  Edward E. Cureton.  The Teacher's Corner: Unbiased Estimation of the Standard Deviation , 1968 .

[36]  Chris Rizos,et al.  Generalised measures of reliability for multiple outliers , 2010 .

[37]  J. P. van Loon,et al.  Robust Estimation and Robust Re-Weighting in Satellite Gravity Modelling , 2008 .

[38]  Bofeng Li,et al.  Efficient Estimation of Variance and Covariance Components: A Case Study for GPS Stochastic Model Evaluation , 2011, IEEE Transactions on Geoscience and Remote Sensing.