A Polynomial Estimation of Measurand Parameters for Samples of Non-Gaussian Symmetrically Distributed Data

  • Zygmunt L. WarszaEmail author
  • Serhii W. Zabolotnii
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 550)


The non-standard method for evaluating of the average and standard deviation of the symmetrically non-Gaussian-distributed data of sample with a priori partial description (unknown PDF) is proposed. This method of statistical estimation is based on the apparatus of stochastic polynomials and uses the higher-order statistics (moment & cumulant description) of random variables. The analytical expressions for finding estimates for the degree of the polynomial s = 3 and their accuracy analyzes are given. It is shown that the uncertainty estimates received for polynomial are generally less than the uncertainty estimates obtained based on the mean (arithmetic average). Reduction factor, which depends on the MSE values of higher order cumulant coefficients, characterizes the degree of the sampling distribution differences from the Gaussian model. The results of statistical modeling, based on the Monte Carlo method, confirmed the effectiveness of the proposed approach are presented.


Estimator Non-Gaussian model Stochastic polynomial Mean value Variance Cumulant coefficients 


  1. 1.
    Supplement 1 to the Guide to the expression of uncertainty in measurement (GUM) – Propagation of distributions using a Monte Carlo method, Guide OIML G 1-101 Ed. 2008Google Scholar
  2. 2.
    Novickij, P.V., Zograf, I.A.: Ocenka pogreshnostiej resultatov izmierenii (Estimation of the measurement result errors). Energoatomizdat, Leningrad (1991). (in Russian)Google Scholar
  3. 3.
    Mendel, J.M.: Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications. Proc. IEEE 79(3), 278–305 (1991)CrossRefGoogle Scholar
  4. 4.
    Kendall, M.G., Stuart, A.: The Advanced Theory of Statistics. Distribution Theory, vol. 1, 3rd edn. Griffin, Spokane Valley (1969)zbMATHGoogle Scholar
  5. 5.
    Toybert, P.: Otsenka tochnosti rezultatov izmereniy (Estimation of accuracy of measurement results). Energoatomizdat, Leningrad (1988). (in Russian)Google Scholar
  6. 6.
    Zaharov, I.P., Klimova, E.A.: Application of excess method to obtain reliable estimate of expanded uncertainty. Systemy obrobky informatsii 3(119), 24–28 (2014). (in Russian)Google Scholar
  7. 7.
    Kuznetsov, B.F., Borodkin, D.K., Lebedeva, L.V.: Cumulant models of additional errors. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie 1(37), 134–138 (2013)Google Scholar
  8. 8.
    De Carlo, L.T.: On the meaning and use of kurtosis. Psychol. Methods 2(3), 292–307 (1997). doi: 10.1037/1082-989X.2.3.292 CrossRefGoogle Scholar
  9. 9.
    Kunchenko, Y.: Polynomial Parameter Estimations of Close to Gaussian Random variables. Shaker Verlag, Aachen (2002)Google Scholar
  10. 10.
    Zabolotnii, S.W., Warsza, Z.L.: Semi-parametric estimation of the change-point of parameters of non-gaussian sequences by polynomial maximization method. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds.) AUTOMATION-2016. AISC, vol. 440, pp. 903–919. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-29357-8_80 CrossRefGoogle Scholar
  11. 11.
    Cramér, H.: Mathematical Methods of Statistics, vol. 9. Princeton University Press, Princeton (1999)zbMATHGoogle Scholar
  12. 12.
    Beregun, V.S., Garmash, O.V., Krasilnikov, A.I.: Mean square error of estimates of cumulative coefficients of the fifth and sixth order. Electron. Model. 36(1), 17–28 (2014)Google Scholar
  13. 13.
    Lilliefors, H.W.: On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J. Am. Stat. Assoc. 62(318), 399–402 (1967)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Industrial Research Institute for Automation and Measurements PIAPWarsawPoland
  2. 2.Cherkasy State Technological UniversityCherkasyUkraine

Personalised recommendations