Abstract
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.
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References
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. 2008
Novickij, P.V., Zograf, I.A.: Ocenka pogreshnostiej resultatov izmierenii (Estimation of the measurement result errors). Energoatomizdat, Leningrad (1991). (in Russian)
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)
Kendall, M.G., Stuart, A.: The Advanced Theory of Statistics. Distribution Theory, vol. 1, 3rd edn. Griffin, Spokane Valley (1969)
Toybert, P.: Otsenka tochnosti rezultatov izmereniy (Estimation of accuracy of measurement results). Energoatomizdat, Leningrad (1988). (in Russian)
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)
Kuznetsov, B.F., Borodkin, D.K., Lebedeva, L.V.: Cumulant models of additional errors. Sovremennye tekhnologii. Sistemnyi analiz. Modelirovanie 1(37), 134–138 (2013)
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
Kunchenko, Y.: Polynomial Parameter Estimations of Close to Gaussian Random variables. Shaker Verlag, Aachen (2002)
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
Cramér, H.: Mathematical Methods of Statistics, vol. 9. Princeton University Press, Princeton (1999)
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)
Lilliefors, H.W.: On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J. Am. Stat. Assoc. 62(318), 399–402 (1967)
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Warsza, Z.L., Zabolotnii, S.W. (2017). A Polynomial Estimation of Measurand Parameters for Samples of Non-Gaussian Symmetrically Distributed Data. In: Szewczyk, R., Zieliński, C., Kaliczyńska, M. (eds) Automation 2017. ICA 2017. Advances in Intelligent Systems and Computing, vol 550. Springer, Cham. https://doi.org/10.1007/978-3-319-54042-9_45
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DOI: https://doi.org/10.1007/978-3-319-54042-9_45
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