Statistical Preliminaries

  • I. Pitas
  • A. N. Venetsanopoulos
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 84)


Many classes of nonlinear filters are based on the field of robust estimation and especially on order statistics. Both of these fields have been developed by statisticians in the last three decades and they have now reached maturity. In this chapter an introduction to robust statistics and order statistics will be given, as a mathematical preliminary to nonlinear filters. The interested reader can find more information in specialized books [1–5].


Order Statistic Maximum Likelihood Estimator Robust Estimation Influence Function Robust Estimator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H.A. David, Order statistics, John Wiley, 1980.Google Scholar
  2. [2]
    P. S. Huber, Robust statistics, John Wiley, 1981.Google Scholar
  3. [3]
    E. L. Lehmann, Theory of point estimation, John Wiley, 1983.Google Scholar
  4. [4]
    J. Hajek, Z. Sidak, Theory of rank tests, Academic Press, 1967.Google Scholar
  5. [5]
    F. Hampel, E. Ronchetti, P. Rousseeuw, W. Stahel, Robust statistics, John Wiley, 1986.Google Scholar
  6. [6]
    B.J. Justusson, “Median filtering: Statistical properties”, in Topics in applied physics, vol. 43, T.S. Huang editor, Springer Verlag, 1981.Google Scholar
  7. [7]
    P. J. Huber, “Robust estimation of a location parameter”, Ann. Math. Statist., vol. 35, pp. 73–101, 1964.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    D. F. Andrews, P. J. Bickel, F. R. Hampel, P. S. Huber, W. H. Rogers, J. W. Tukey, Robust estimates of location: Survey and advances, Princeton University Press, 1972.Google Scholar
  9. [9]
    P. J. Bickel, “One step Huber estimates in the linear model”, J. Am. Statist. Assoc., vol. 70, pp. 428–434.Google Scholar
  10. [10]
    R. B. Murphy, On test for outlying observations, Ph.D. Thesis, Princeton University, 1951.Google Scholar
  11. [11]
    D. M. Hawkins, “Fractiles of an extented multiple outlier test”, J. Statist. Comput. Simulation, vol. 8, pp. 227–236, 1979.CrossRefGoogle Scholar
  12. [12]
    V. Barnett, T. Lewis, Outliers in statistical data, Wiley, 1978.Google Scholar
  13. [13]
    J. L. Hodges Jr., E. L. Lehmann, “Estimates of location based on rank tests”, Ann. Math. Statist., vol. 34, pp. 598–611, 1963.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    L. A. Jaeckel, Robust estimates of location, Ph.D. Thesis, University of California, Berkeley, 1969.Google Scholar
  15. [15]
    D. A. Lax, An interim report of a Monte Carlo study of robust estimators of width, Technical report 93, series 2, Department of Statistics, Princeton University, 1975.Google Scholar
  16. [16]
    J. Wolfowitz, “The minimum distance model”, Ann. Math. Statist. vol. 28, pp. 75–88, 1957.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    M. V. Johns, “Robust Pitman-like estimators”, in Robustness in statistics, R. L. Launer and G. N. Wilkinson editors, Academic Press, 1979.Google Scholar
  18. [18]
    P. J. Rousseeuw, V. Yokai, “Robust regression by means of S-estimators”, in Robust and nonlinear time series analysis, J. Franke, W. Hardie, R. D. Martin editors, Lecture notes in statistics, vol.26, Springer, 1984.Google Scholar
  19. [19]
    J. W. Tukey, Exploratory data analysis, Addison-Wesley, 1970, 1977.Google Scholar
  20. [20]
    P. Papantoni-Kazakos, R.M. Gray, “Robustness of estimators on stationary observations”, Ann. Prob., vol. 7, pp. 989–1002, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    B. Kleiner, R.D. Martin, D.J. Thomson, “Robust estimation of power spectra”, J. Roy. Statist. Soc. Ser. B, vol. 41, pp. 313–351, 1979.MathSciNetzbMATHGoogle Scholar
  22. [22]
    R.D. Martin, D.J. Thomson, “Robust resistant spectrum estimation”, Proceedings of IEEE, vol. 70, pp. 1097–1115, Sept. 1982.CrossRefGoogle Scholar
  23. [23]
    B.T. Poljak, Y.Z.Tsypkin, “Robust identification”, Automatica, vol. 16, pp. 53–63, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    S.A Kassam, H.V. Poor, “Robust techniques for signal processing: A survey”, Proceedings of IEEE, vol. 73, no. 3, pp. 433–481, March 1985.zbMATHCrossRefGoogle Scholar
  25. [25]
    H.V. Poor, “On robust Wiener filtering”, IEEE Transactions on Automatic Control, vol. AC-25, pp. 531–536, June 1980.Google Scholar
  26. [26]
    K.S. Vastola, H.V. Poor, “Robust Wiener-Kolmogoroff theory”, IEEE Transactions on Information Theory, vol. IT-30, pp. 316–327, March 1984.Google Scholar
  27. [27]
    S.A. Kassam, “Robust hypothesis testing and robust time series interpolation and regression”, Journal of Time Series Analysis, vol. 3, pp. 185–194, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    K.M. Ahmed, R.J. Evans, “Robust signal and array processing”, Proceedings of IEE, vol. 129, Pt. F, no. 4, pp. 297–302, Aug. 1982.Google Scholar
  29. [29]
    A.H. El-Sawy, V.D. VandeLinde, “Robust detection of known signals”, IEEE Transactions on Information Theory, vol. IT-23, pp. 722–727, Nov. 1977.Google Scholar
  30. [30]
    J.W. Modestino, “Adaptive detection of signals in impulsive noise environments”, IEEE Transactions on Communications, vol. COM-25, pp. 1022–1027, Sept. 1977.Google Scholar
  31. [31]
    C.G. Boncelet, “Algorithms to compute order statistic distributions”, SIAM J. Stat. Comput., vol. 8, no. 5, pp. 868–876, Sept. 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    M.V. Johns, “Robust Pitman-like estimators”, in Robustness in statistics, R.L. Launer and G.N. Wilkinson editors, Academic Press, 1979.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • I. Pitas
    • 1
  • A. N. Venetsanopoulos
    • 2
  1. 1.Aristotelian University of ThessalonikiGreece
  2. 2.University of TorontoCanada

Personalised recommendations