Skip to main content
Log in

Observation error model selection by information criteria vs. normality testing

Studia Geophysica et Geodaetica Aims and scope Submit manuscript

Abstract

To extract the best possible information from geodetic and geophysical observations, it is necessary to select a model of the observation errors, mostly the family of Gaussian normal distributions. However, there are alternatives, typically chosen in the framework of robust M-estimation. We give a synopsis of well-known and less well-known models for observation errors and propose to select a model based on information criteria. In this contribution, we compare the Akaike information criterion (AIC) and the Anderson-Darling (AD) test and apply them to the test problem of fitting a straight line. The comparison is facilitated by a Monte Carlo approach. It turns out that the model selection by AIC has some advantages over the AD test.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Anderson T.W. and Darling D.A., 1952. Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes. Ann. Math. Stat., 23, 193–212, DOI:10.1214/aoms /1177729437.

    Article  Google Scholar 

  • Anderson T.W. and Darling D.A., 1954. A Test of goodness-of-fit. J. Amer. Stat. Assoc., 49, 765–769, DOI: 10.2307/2281537.

    Article  Google Scholar 

  • Akaike H., 1974. A new look at the statistical model identification. IEEE Trans. Autom. Control, 19, 716–723.

    Article  Google Scholar 

  • Burnham K.P. and Anderson D.R., 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer-Verlag, Berlin, Germany, DOI: 10.1007/b97636.

    Google Scholar 

  • Cai J., Grafarend E. and Hu C., 2007. The statistical property of the GNSS carrier phase observations and its effects on the hypothesis testing of the related estimators. In: Proceedings of the 20th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2007). The Institute of Navigation, Manassas, VA, 331–338.

    Google Scholar 

  • Cramér H., 1928. On the composition of elementary errors. Skandinavisk Aktuarietidskrift, 11, 13-74, 141–180.

    Google Scholar 

  • D’Agostino R.B., 1970. Transformation to normality of the null distribution of g1. Biometrika, 57, 679–681, DOI: 10.1093/biomet/57.3.679.

    Google Scholar 

  • Hampel F.R., 2001. Robust statistics: A brief introduction and overview. In: Carosio A. and Kutterer H. (Eds). First International Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GIS. Institute of Geodesy and Photogrametry, ETH Zurich, Zurich, Switzerland.

    Google Scholar 

  • Huber P.J., 1964. Robust estimation of a location parameter. Ann. Stat., 53, 73–101.

    Article  Google Scholar 

  • Huber P.J., 2009. Robust Statistics. 2nd Edition. John Wiley & Sons Inc, New York. ISBN 978-0- 470-12990-6.

    Book  Google Scholar 

  • Jarque C.M. and Bera A.K., 1980. Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Econ. Lett., 6, 255–259, DOI: 10.1016/0165- 1765(80)90024-5.

    Article  Google Scholar 

  • Klees R., Ditmar P. and Broersen P., 2002. How to handle colored observation noise in large leastsquares problems. J. Geodesy, 76, 629–640, DOI: 10.1007/s00190-01-0392-4.

    Article  Google Scholar 

  • Kolmogorov A., 1933. Sulla determinazione empirica di una legge di distribuzione. Giornale dell’Istituto Italiano degli Attuari, 4, 83–91 (in Italian).

    Google Scholar 

  • Kutterer H., 2001. Uncertainty assessment in geodetic data analysis. In: Carosio A. and Kutterer H. (Eds). First International Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GIS. Institute of Geodesy and Photogrametry, ETH Zurich, Zurich, Switzerland.

    Google Scholar 

  • Lehmann R., 2012. Geodätische Fehlerrechnung mit der skalenkontaminierten Normalverteilung. Allgemeine Vermessungs-Nachrichten, 5/2012. VDE-Verlag Offenbach (in German).

    Google Scholar 

  • Lehmann R., 2013. On the formulation of the alternative hypothesis for geodetic outlier detection. J. Geodesy, 87, 373–386.

    Article  Google Scholar 

  • Lehmann R., 2014. Transformation model selection by multiple hypothesis testing. J. Geodesy, 88, 1117–1130, DOI: 10.1007/s00190-014-0747-3.

    Article  Google Scholar 

  • Lilliefors H.W., 1967. On the Kolmogorov-Smirnov for normality with mean and variance unknown. J. Am. Stat. Assoc., 62, 399–402.

    Article  Google Scholar 

  • Luo X., 2013. GPS Stochastic Modelling - Signal Quality Measures and ARMA Processes. Springer-Verlag, Berlin, Heidelberg, Germany, DOI: 10.1007/978-3-642-34836-5.

    Book  Google Scholar 

  • Luo X., Mayer M. and Heck B., 2011. On the probability distribution of GNSS carrier phase observations. GPS Solut., 15, 369–379, DOI: 10.1007/s10291-010-0196-2.

    Article  Google Scholar 

  • Miller R.G., 1981. Simultaneous Statistical Inference. Springer, New York. ISBN:0-387-90548-0.

    Book  Google Scholar 

  • Pearson K., 1900. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Phil. Mag. Ser. 5, 50(302), 157–175, DOI: 10.1080/14786440009463897.

    Article  Google Scholar 

  • Razali N.M. and Wah Y.B., 2011. Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. J. Stat. Model. Anal., 2, 21–33.

    Google Scholar 

  • Shapiro S.S. and Francia R.S., 1972. An approximate analysis of variance test for normality. J. Am. Stat. Assoc., 67, 215–216, DOI: 10.1080/01621459.1972.10481232.

    Article  Google Scholar 

  • Shapiro S.S. and Wilk M.B., 1965. An analysis of variance test for normality (complete samples). Biometrica, 52, 591–611.

    Article  Google Scholar 

  • Smirnov N., 1948. Table for estimating the goodness of fit of empirical distributions. Ann. Math. Stat., 19, 279–281, DOI: 10.1214/aoms/1177730256.

    Article  Google Scholar 

  • Stephens M.A., 1974. EDF statistics for goodness of fit and some comparisons. J. Amer. Stat. Assoc., 69, 730–737, DOI: 10.2307/2286009.

    Article  Google Scholar 

  • Tanizaki H., 2004. Computational Methods in Statistics and Econometrics. Marcel Dekker, New York, ISBN-13: 978–0824748043.

    Book  Google Scholar 

  • Teunissen P.J.G., 2000. Testing Theory: an Introduction. 2nd Edition. Series on Mathematical Geodesy and Positioning, Delft University of Technology, Delft, The Netherlands.

    Google Scholar 

  • Tiberius C.C.J.M. and Borre K., 2000. Are GPS data normally distributed. In: Schwarz K.P. (Ed.), Geodesy Beyond 2000. International Association of Geodesy Symposia 121. pringer-Verlag, Heidelberg, Germany, 243–248.

    Chapter  Google Scholar 

  • Tukey J.W., 1960. A survey of sampling from contaminated distributions. In: Olkin I. (Ed.), Contributions to Probability and Statistics. University Press Stanford California, Stanford, CA.

    Google Scholar 

  • Verhagen S. and Teunissen P.J.G., 2005. On the probability density function of the GNSS ambiguity residuals. GPS Solut., 10, 21–28, DOI: 10.1007/s10291-005-0148-4.

    Article  Google Scholar 

  • von Mises R., 1931. Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und Theoretischen Physik. F. Deutike, Leipzig, Germany (in German).

    Google Scholar 

  • Wisniewski Z., 2014. M-estimation with probabilistic models of geodetic observations. J. Geodesy, 88, 941–957, DOI: 10.1007/s00190-014-0735-7.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rüdiger Lehmann.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lehmann, R. Observation error model selection by information criteria vs. normality testing. Stud Geophys Geod 59, 489–504 (2015). https://doi.org/10.1007/s11200-015-0725-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11200-015-0725-0

Keywords

Navigation