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Conference on Automation

AUTOMATION 2019: Automation 2019 pp 636–649Cite as

Estimation of Linear Regression Parameters of Symmetric Non-Gaussian Errors by Polynomial Maximization Method

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Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 920))

Abstract

In this paper, a new way of estimation of single-factor linear regression parameters of symmetrically distributed non-Gaussian errors is proposed. This new approach is based on the Polynomial Maximization Method (PMM) and uses the description of random variables by higher order statistics (moments and cumulants). Analytic expressions that allow to find estimates and analyze their asymptotic accuracy are obtained for the degree of polynomial S = 3. It is shown that the variance of polynomial estimates can be less than the variance of estimates of the ordinary least squares’ method. The increase of accuracy depends on the values of cumulant coefficients of higher order of the random regression errors. The statistical modeling of the Monte Carlo method has been performed. The results confirm the effectiveness of the proposed approach.

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References

  1. Anscombe, F.J.: Topics in the investigation of linear relations fitted by the method of least squares. J. R. Stat. Soc. Ser. B (Methodological) 29, 1–52 (1967)

    MathSciNet  MATH  Google Scholar 

  2. Cox, D.R., Hinkley, D.V.: A note on the efficiency of least-squares estimates. J. R. Stat. Soc. Ser. B (Methodological) 30, 284–289 (1968)

    MathSciNet  MATH  Google Scholar 

  3. Schechtman, E., Schechtman, G.: Estimating the parameters in regression with uniformly distributed errors. J. Stat. Comput. Simul. 26(3–4), 269–281 (1986). https://doi.org/10.1080/00949658608810965

    Article  MathSciNet  MATH  Google Scholar 

  4. Galea, M., Paula, G.A., Bolfarine, H.: Local influence in elliptical linear regression models. J. R. Stat. Soc. Ser. D: Stat. 46(1), 71–79 (1997)

    Article  Google Scholar 

  5. Liu, S.: Local influence in multivariate elliptical linear regression models. Linear Algebra Appl. 354(1–3), 159–174 (2002). https://doi.org/10.1016/S0024-3795(01)00585-7

    Article  MathSciNet  MATH  Google Scholar 

  6. Ganguly, S.S.: Robust regression analysis for non-normal situations under symmetric distributions arising in medical research. J. Modern Appl. Stat. Meth. 13(1), 446–462 (2014). https://doi.org/10.22237/jmasm/1398918480

    Article  Google Scholar 

  7. Zeckhauser, R., Thompson, M.: Linear regression with non-normal error terms. Rev. Econ. Stat. 52(3), 280–286 (1970)

    Article  Google Scholar 

  8. Bartolucci, F., Scaccia, L.: The use of mixtures for dealing with non-normal regression errors. Comput. Stat. Data Anal. 48(4), 821–834 (2005). https://doi.org/10.1016/j.csda.2004.04.005

    Article  MathSciNet  MATH  Google Scholar 

  9. Seo, B., Noh, J., Lee, T., Yoon, Y.J.: Adaptive robust regression with continuous Gaussian scale mixture errors. J. Korean Stat. Soc. 46(1), 113–125 (2017). https://doi.org/10.1016/j.jkss.2016.08.002

    Article  MathSciNet  MATH  Google Scholar 

  10. Tiku, M.L., Islam, M.Q., Selçuk, A.S.: Non-normal regression II. Symmetric distributions. Commun. Stat. Theory Meth. 30(6), 1021–1045 (2001). https://doi.org/10.1081/STA-100104348

    Article  MATH  Google Scholar 

  11. Andargie, A.A., Rao, K.S.: Estimation of a linear model with two-parameter symmetric platykurtic distributed errors. J. Uncertaint. Anal. Appl. 1(1), 1–19 (2013)

    Article  Google Scholar 

  12. Atsedeweyn, A.A., Srinivasa Rao, K.: Linear regression model with generalized new symmetric error distribution. Math. Theory Model. 4(2), 48–73 (2014). https://doi.org/10.1080/02664763.2013.839638

    Article  Google Scholar 

  13. Huber, P.J., Ronchetti, E.M.: Robust Statistics. Wiley, Hoboken (2009). https://doi.org/10.1002/9780470434697

    Book  MATH  Google Scholar 

  14. Narula, S.C., Wellington, J.F.: The minimum sum of absolute errors regression: a state of the art survey. Int. Stat. Rev. 50(3), 317–326 (1982)

    Article  MathSciNet  Google Scholar 

  15. Koenker, R., Hallock, K.: Quantile regression: an introduction. J. Economic. Perspect. 15(4), 43–56 (2001)

    Article  Google Scholar 

  16. Tarassenko, P.F., Tarima, S.S., Zhuravlev, A.V., Singh, S.: On sign-based regression quantiles. J. Stat. Comput. Simul. 85(7), 1420–1441 (2015). https://doi.org/10.1080/00949655.2013.875176

    Article  MathSciNet  Google Scholar 

  17. Dagenais, M.G., Dagenais, D.L.: Higher moment estimators for linear regression models with errors in the variables. J. Econom. 76(1–2), 193–221 (1997). https://doi.org/10.1016/0304-4076(95)01789-5

    Article  MathSciNet  MATH  Google Scholar 

  18. Cragg, J.G.: Using higher moments to estimate the simple errors-in-variables model. RAND J. Econ. 28, S71 (1997). https://doi.org/10.2307/3087456

    Article  Google Scholar 

  19. Gillard, J.: Method of moments estimation in linear regression with errors in both variables. Commun. Stat. Theory Meth. 43(15), 3208–3222 (2014)

    Article  MathSciNet  Google Scholar 

  20. Zabolotnii, S., Warsza, Z., Tkachenko, O.: Polynomial estimation of linear regression parameters for the asymmetric pdf of errors. In: Advances in Intelligent Systems and Computing. vol. 743, pp. 758–772. Springer (2018). https://doi.org/10.1007/978-3-319-77179-3_75

    Google Scholar 

  21. Kunchenko, Y.: Polynomial Parameter Estimations of Close to Gaussian Random variables. Shaker Verlag, Aachen (2002)

    Google Scholar 

  22. Warsza, Z.L., Zabolotnii, S.W.: A polynomial estimation of measurand parameters for samples of non-Gaussian symmetrically distributed data. In: Advances in Intelligent Systems and Computing, vol. 550, pp. 468–480. Springer (2017). http://doi.org/10.1007/978-3-319-54042-9_45

    Chapter  Google Scholar 

  23. Warsza, Z.L., Zabolotnii, S.W.: Uncertainty of measuring data with trapeze distribution evaluated by the polynomial maximization method. Przemysł Chemiczny 1(12), 68–71 (2017). https://doi.org/10.15199/62.2017.12.6. (in Polish)

    Article  Google Scholar 

  24. Warsza, Z., Zabolotnii, S.: Estimation of measurand parameters for data from asymmetric distributions by polynomial maximization method. In: Advances in Intelligent Systems and Computing, vol. 743, pp. 746–757. Springer (2018). https://doi.org/10.1007/978-3-319-77179-3_74

    Google Scholar 

  25. Zabolotnii, S.W., Warszam, Z.L.: Semi-parametric estimation of the change-point of parameters of non-Gaussian sequences by polynomial maximization method. In: Advances in Intelligent Systems and Computing, vol. 440, pp. 903–919. Springer (2016). http://doi.org/10.1007/978-3-319-29357-8_80

    Chapter  Google Scholar 

  26. Palahin, V., Juh, J.: Joint signal parameter estimation in non–Gaussian noise by the method of polynomial maximization. J. Electr. Eng. 67, 217–221 (2016). https://doi.org/10.1515/jee-2016-0031

    Article  Google Scholar 

  27. Cramér, H.: Mathematical Methods of Statistics, vol. 9. Princeton University Press, Princeton (2016)

    MATH  Google Scholar 

  28. Cook, R.D., Weisberg, S.: Residuals and Influence in Regression. Monographs on Statistics and Applied Probability. Chapman and Hall, New York (1982). https://doi.org/10.2307/1269506

    Book  MATH  Google Scholar 

  29. Stone, C.J.: Adaptive maximum likelihood estimators of a location parameter. Annal. Stat. 3(2), 267–284 (1975). https://doi.org/10.1214/aos/1176343056

    Article  MathSciNet  MATH  Google Scholar 

  30. Boos, D.D.: Detecting skewed errors from regression residuals. Technometrics 29(1), 83–90 (1987). https://doi.org/10.1080/00401706.1987.10488185

    Article  MathSciNet  MATH  Google Scholar 

  31. Jarque, C.M., Bera, A.K.: A test for normality of observations and regression residuals. Int. Stat. Rev. 55(2), 163–172 (2012)

    Article  MathSciNet  Google Scholar 

  32. Keeling, C.D., Whorf, T.P.: Scripps Institution of Oceanography (SIO). University of California, La Jolla, California USA 92093-0220. ftp://cdiac.esd.ornl.gov/pub/maunaloa-co2/maunaloa.co2

    Google Scholar 

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Author information

Authors and Affiliations

  1. Cherkasy State Technological University, Cherkasy, Ukraine

    Serhii W. Zabolotnii & Oleksandr Tkachenko

  2. Industrial Research Institute for Automation and Measurements PIAP, Al. Jerozolimskie 202, 02-486, Warsaw, Poland

    Zygmunt L. Warsza

Authors
  1. Serhii W. Zabolotnii
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  2. Zygmunt L. Warsza
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  3. Oleksandr Tkachenko
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Corresponding author

Correspondence to Zygmunt L. Warsza .

Editor information

Editors and Affiliations

  1. Industrial Research Institute for Automation and Measurements PIAP Al, Warsaw, Poland

    Roman Szewczyk

  2. Industrial Research Institute for Automation and Measurements PIAP Al, Warsaw, Poland

    Cezary Zieliński

  3. Industrial Research Institute for Automation and Measurements PIAP Al, Warsaw, Poland

    Małgorzata Kaliczyńska

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Zabolotnii, S.W., Warsza, Z.L., Tkachenko, O. (2020). Estimation of Linear Regression Parameters of Symmetric Non-Gaussian Errors by Polynomial Maximization Method. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds) Automation 2019. AUTOMATION 2019. Advances in Intelligent Systems and Computing, vol 920. Springer, Cham. https://doi.org/10.1007/978-3-030-13273-6_59

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