Skip to main content
SpringerLink
Log in

Robust estimation of a correlation coefficient for ε-contaminated bivariate normal distributions

  • Stochastic Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

An Erratum to this article was published on 01 February 2007

Abstract

Robust estimations of a correlation coefficient, based on: (i) direct robust analogues of a sample correlation coefficient, (ii) nonparametric estimations of correlation, (iii) robust regression, (iv) robust estimation of the major constituents of a variance, (v) stable estimation of parameters, and (vi) preliminary removal of outliers from data with the following application of a sample correlation coefficient to the residuary observations, are studied. Their application to contaminated normal models on small and large samples is studied, the best robust estimations from the suggested are pointed out.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Huber, P.J., Robust Statistics, New York: Wiley, 1981. Translated under the title Robastnost’v statistike, Moscow: Mir, 1984.

    Book  MATH  Google Scholar 

  2. Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., and Stahel, W.A., Robust Statistics: The Approach Based on Influence Functions, New York: Wiley, 1986. Translated under the title Robastnost’v statistike. Podkhod na osnove funktsii vliyaniya, Moscow: Mir, 1989.

    MATH  Google Scholar 

  3. Rogers, J.L. and Nicewander, W.A., Thirteen Ways to Look at the Correlation Coefficient, Am. Stat., 1988, vol. 42, pp. 59–66.

    Article  Google Scholar 

  4. Rovine, M.J. and von Eye, A., A 14th Way to Look at a Correlation Coefficient: Correlation As the Proportion of Matches, Am. Stat., 1997, vol. 51, pp. 42–46.

    Article  Google Scholar 

  5. Spearman, C., The Proof and Measurement of Association between Two Things, Am. J. Psychol., 1904, vol. 15, pp. 88–93.

    Google Scholar 

  6. Blomqvist, N., On a Measure of Dependence between Two Random Variables, Ann. Math. Statist., 1950, vol. 21, pp. 593–600.

    MathSciNet  MATH  Google Scholar 

  7. Kendall, M.G., Rank and Product-Moment Correlation, Biometrika, 1949, vol. 36, pp. 177–180.

    Article  MathSciNet  MATH  Google Scholar 

  8. Tukey, J.W., A Survey of Sampling from Contaminated Distributions, in Contrib. Prob. Statist., Olkin, I., Ed., Stanford: Stanford Univ. Press., 1960, pp. 448–485.

    Google Scholar 

  9. Gnanadesikan, R. and Kettenring, J.R., Robust Estimates, Residuals and Outlier Detection with Multiresponse Data, Biometrics, 1972, vol. 28, pp. 81–124.

    Article  Google Scholar 

  10. Devlin, S.J., Gnanadesikan, R., and Kettenring, J.R., Robust Estimation and Outlier Detection with Correlation Coefficient, Biometrika, 1975, vol. 62, pp. 531–545.

    Article  MATH  Google Scholar 

  11. Huber, P.J., Robust Estimation of a Location Parameter, Ann. Math. Stat., 1964, vol. 35, pp. 73–101.

    MathSciNet  MATH  Google Scholar 

  12. Hampel, F.R., Contributions to the Theory of Robust Estimation, PhD Dissertation, Berkeley: Univ. of California, 1968.

    Google Scholar 

  13. Box, G.E.P., Non-Normality and Test on Variances, Biometrika, 1953, vol. 40, pp. 318–335.

    Article  MathSciNet  MATH  Google Scholar 

  14. Kolmogorov, A.N., The Studies in the Field of Limit Theorems of Probability Theory during the Last Several Years, Bull. Mosk. Gos. Univ., 1953, vol. 10, pp. 28–39.

    Google Scholar 

  15. Prokhorov, Yu.V., Covergence of Random Prossesses and Limit Theorems of Probability Theory, Teor. Veroyatn. Primenen., 1956, vol. 1, pp. 157–214.

    Google Scholar 

  16. Shulenin, V.P., Vvedenie v robastnuyu statistiku (Introduction to Robust Statistics), Tomsk: Tomsk. Univ., 1993.

    Google Scholar 

  17. Bickel, P.J., Another Look at Robustness: A Review of Reviews and Some New Developments, Scand. J. Stat. Theory Appl., 1976, vol. 3, pp. 145–168.

    MathSciNet  MATH  Google Scholar 

  18. Polyak, B.T. and Tsypkin, Ya.Z., Robust Identification, Automatika, 1980, vol. 16, pp. 53–65.

    Article  MathSciNet  Google Scholar 

  19. Hampel, F.R., A General Qualitative Fefinition of Robustness, Ann. Math. Stat., 1971, vol. 42, pp. 1887–1896.

    MathSciNet  MATH  Google Scholar 

  20. Hampel, F.R., The Influence Curve and Its Role in Robust Estimation, L. Am. Stat. Assoc., 1974, vol. 69, pp. 383–393.

    Article  MathSciNet  MATH  Google Scholar 

  21. Pasman, V.P. and Shevlyakov, G.L., Robust Methods of the Estimation of a Correlation Coefficient, Avtom. Telemekh., 1987, no. 3, pp. 70–80.

  22. Shevlyakov, G.L. and Vilchevski, N.O., Robustness in Data Analysis: Criteria and Methods, Utrecht: VSP, 2002.

    Google Scholar 

  23. Shurygin, A.M., Variational Optimization of the Estimator Stability, Avtom. Telemekh., 1994, no. 11, pp. 73–86.

  24. Sgurygin, A.M., Prikladnaya stokhastika: robastnost’, otsenivanie i prognozirovanie (Applied Stochastics: Robustness, Estimation, and Prediction), Moscow: Finansy i Statistika, 2000.

    Google Scholar 

  25. Hampel, F.R., Rousseeuw, P.J., and Ronchetti, E., The Change-of-Variance Curve and Optimal Redescending M-estimators, J. Am. Stat. Assoc., 1981, vol. 76, pp. 643–648.

    Article  MathSciNet  MATH  Google Scholar 

  26. Meshalkin, L.D., Some Mathematical Methods for the Study of Non-Communicable Diseases, Proc. 6th Int. Meeting of Uses of Epidemol. in Planning Health Services, Primosten, Yugoslavia, 1971, vol. 1, pp. 250–256.

    Google Scholar 

  27. Aivazian, S.A., Bukhshtaber, V.M., Enyukov, I.S., and Meshalkin, L.D., Prikladnaya statistika. Klassifikatsiya i snizhenie razmernosti (Applied Statistics: Classification and Dimensionality Reduction), Moscow: Finansy i Statistika, 1989.

    Google Scholar 

  28. Jaynes, E.T., Probability Theory. The Logic of Science, Cambridge: Cambridge Univ. Press, 2003.

    MATH  Google Scholar 

  29. Kazakos, D. and Papantoni-Kazakos, P., Detection and Estimation, Rockville: Computer Science Press, 1990.

    Google Scholar 

  30. Delic, H. and Hocanin, A., Successive Interference Cancellation Using Robust Correlation, Proc. 1st IEEE Balcan Conf. Signal Proc., Commun., Circuitc, Syst., Istanbul, 2000, pp. 1221–1224.

  31. Kim, J. and Fessler, J.A., Intensity-Based Image Registration Using Robust Correlation Coefficients, IEEE Trans. Medical Imaging, 2004, vol. 23, pp. 1430–1443.

    Article  Google Scholar 

  32. Trujillo, M. and Izquierdo, E., A Robust Correlation Measure for Correspondence Estimation, Proc. 2nd Int. Sympos. on 3D Data Processing, Visualization and Transmission, Thessaloniki, 2004.

  33. Jurechkova, J., Statistical Tests on Tail Index of a Probability Distribution, METRON—Int. J. Stat., 2003, vol. 61, no. 2, pp. 151–190.

    Google Scholar 

  34. Andrews, D.F., Bickel, P.J., Hampel, F.R., et al., Robust Estimates of Location, Princeton: Princeton Univ. Press, 1972.

    MATH  Google Scholar 

  35. Hawkins, D.M., The Identification of Outliers, London: Chapman & Hall, 1980.

    Google Scholar 

  36. Rousseeuw, P.J. and Leroy, A.M., Robust Regression and Outlier Detection, New York: Wiley, 1987.

    Book  MATH  Google Scholar 

  37. Falk, M., A Note on the Correlation Mediam for Elliptical distributions, J. Multivar. Anal., 1998, vol. 67, pp. 306–317.

    Article  MathSciNet  MATH  Google Scholar 

  38. Kendall, M.G. and Stuart, A., Statistical Inference and Relationship, London: Griffin, 1961. Translated under the title Statisticheskie vyvody i svyazi, Moscow: Mir, 1973.

    Google Scholar 

  39. Mudrov, V.I. and Kushko, V.L., Metody obrabotki izmerenii: kvazipravdopodobnye otsenki (Data Processing Methods: Quasiplausible Estimations), Moscow: Radio i Svyaz’, 1983.

    Google Scholar 

  40. Rousseeuw, P.J., Least Median of Squares Regression, J. Am. Stat. Assoc., 1984, vol. 79, pp. 871–880.

    Article  MathSciNet  MATH  Google Scholar 

  41. Shevlyakov, G.L. and Vilchevsky, N.O., Minimax Variance Estimation of a Correlation Coefficient for Epsilon-contaminated Bivariate Normal Distributions, Stat. Prob. Lett., 2002, vol. 57, pp. 91–100.

    Article  MATH  Google Scholar 

  42. Barnett, V. and Lewis, T., Outliers in Statistical Data, New York: Wiley, 1978.

    MATH  Google Scholar 

  43. Atkinson, A. and Riani, M., Robust Diagnostics Regression Analysis, New York: Springer, 2000.

    Google Scholar 

  44. Rocke, D.M. and Woodruff, D.L., Identification of Outliers in Multivariate Data, J. Am. Stat. Assoc., 1996, vol. 91, pp. 1047–1061.

    Article  MathSciNet  MATH  Google Scholar 

  45. Rousseeuw, P.J., Multivariate Estimation with High Breakdown Point, in Math. Stat. Applic., Grossman, W. et al., Ed., Reidel, Dodrecht, 1985, pp. 283–297.

    Google Scholar 

  46. Rocke, D.M. and Woodruff, D.L., Computation of Robust Estimates of Multivariate Location and Shape, Statistica Neerlandica, 1993, vol. 47, pp. 27–42.

    Article  MathSciNet  Google Scholar 

  47. Shevlyakov, G.L. and Khvatova, T.Yu., On Robust Estimation of a Correlation Coefficient and Correlation Matrix, MODA 5—Advances in Model-Oriented Data Analysis, Atkinson, A.C. et al, Eds., Heidelberg: Physica, 1998, pp. 153–162.

    Google Scholar 

  48. Knuth, D.E., The Art of Computer Programming, Fundamental Algorithms, New York: Addison-Wesley, 1975, vol. 1. Translated under the title Iskusstvo programmirovaniya. Fundamental’nye algoritmy, Moscow: Vil’yams, 2000.

    Google Scholar 

  49. Weiszfeld, E., Sur le point par leque la somme des distansis de n points donnes est minimum, Tôhoku Math. J., 1937, vol. 37, pp. 355–386.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Technology, Kumi, South Korea

    Zh. V. Li

  2. Institute of Science and Technology, Gwangju, South Korea

    G. L. Shevlyakov & V. I. Shin

Authors
  1. Zh. V. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. L. Shevlyakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. I. Shin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © Zh.V. Li, G.L. Shevlyakov, V.I. Shin, 2006, published in Avtomatika i Telemekhanika, 2006, No. 12, pp. 86–105.

This work was supported by the Research Foundation of the Institute of Technology, Kumi, South Korea.

An erratum to this article is available at http://dx.doi.org/10.1134/S0005117907020178.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Z.V., Shevlyakov, G.L. & Shin, V.I. Robust estimation of a correlation coefficient for ε-contaminated bivariate normal distributions. Autom Remote Control 67, 1940–1957 (2006). https://doi.org/10.1134/S0005117906120071

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117906120071

PACS number

Search

Navigation