2. Home
  3. ARCHIVE
  4. 2023
  5. VOL 45, NO 5 (2023)
  6. pp. 20-38

CLASSIFICATION OF MODELS OF TWO-COMPONENT MIXTURES OF SYMMETRICAL DISTRIBUTIONS WITH ZERO KURTOSIS COEFFICIENT

A.I. Krasilnikov

Èlektron. model. 2023, 45(5):20-38

https://doi.org/10.15407/emodel.45.05.020

ABSTRACT

On the basis of a family of two-component mixtures of distributions, a class K of symmetric non-Gaussian distributions with a zero kurtosis coefficient is defined, which is divided into two groups and five types. The dependence of the fourth-order cumulant on the weight coefficient of the mixture is studied, as a result of which the conditions are determined under which the kurtosis coefficient of the mixture is equal to zero. The use of a two-component mixture of Subbotin distributions for modeling single-vertex symmetric distributions with a zero kurtosis coefficient is justified. Examples of symmetric non-Gaussian distributions with zero kurtosis coefficient are given. The use of class K models gives a practical opportunity at the design stage to compare the effectiveness of the developed methods and systems for non-Gaussian signals with zero coefficients of asymmetry and kurtosis processing.

KEYWORDS

non-Gaussian distributions, two-component mixtures of distributions, cumulative coefficients, skewness coefficient, kurtosis coefficient.

REFERENCES

  1. Artyushenko, V.M. and Volovach, V.I. (2017), “Identification of distribution parameters of additive and multiplicative non-Gaussian interference”, Avtometriya, Vol. 53, no 3, pp. 33-
    https://doi.org/10.3103/S8756699017030050
  2. Beregun,S., Gorovetska, T.A. and Krasylnіkov, O.І. (2011), “The statistical analysis of noise of knee joints”, Akustichnyi vіsnyk, Vol. 14, no. 2, pp. 3-15.
  3. Zapevalov, A.S. and Garmashov, A.V. (2021), “Skewness and Kurtosis of the Surface Wa­ve in the Coastal Zone of the Black Sea”, Morskoy gidrofizicheskiy zhurnal, Vol. 37, no. 4, 447-459. 
    https://doi.org/10.22449/0233-7584-2021-4-447-459
  4. Krasilnikov, A.I., Beregun, V.S. and Polobyuk, T.A. (2019), Kumulyantnye metody v zada­chakh shumovoy diagnostiki teploenergeticheskogo oborudovaniya [Cumulant methods in the problems of noise diagnostics of heat-and-power equipment], Osvita Ukrainy, Kyiv, Ukraine.
  5. Kuznetsov, B.F., Borodkin, D.K., Lebedeva, L.V. (2013), “Cumulant Models of Additio­nal Errors”, Sovremennye tekhnologii. Sistemnyy analiz. Modelirovanie, 1 (37), pp. 134-138.
  6. Malkin, A.L., Sorin, A.Ya. and Finikov, D.B. (1986), “Application of Cumulant Analysis in Statistical Processing of Seismic Records”, Geologiya i geofizika, no. 5, pp. 75-
  7. Novitskii, P.V. and Zograf, I.A. (1991), Otsenka pogreshnostei rezultatov izmerenii [Error Estimation in Measurement Results], Energoatomizdat, St. Petersburg, Russia.
  8. Arnau, J., Bendayan, R., Blanca, M.J. and Bono R. (2013), “The effect of skewness and kurtosis on the robustness of linear mixed models”, Behavior Research Methods, Vol. 45, no. 3, pp. 873–879.
    https://doi.org/10.3758/s13428-012-0306-x
  9. Blanca, M.J., Arnau, J., Lopez-Montiel, D., Bono, R. and Bendayan R. (2013), “Skewness and kurtosis in real data samples”, Methodology, no. 9, pp. 78-
    https://doi.org/10.1027/1614-2241/a000057
  10. De Carlo, L.T. (1997), “On the meaning and use of kurtosis”, Psychological Methods, Vol. 2, no. 3, pp. 292-307.
    https://doi.org/10.1037/1082-989X.2.3.292
  11. Downey, T.J.G., Martin, P., Sedlaček, M. and Beaulieu, L.Y. (2017), “A Computational Analysis of the Application of Skewness and Kurtosis to Corrugated and Abraded Surfa­ces”, Quarterly Physics Review, Vol. 3, Issue 3, pp. 1-
  12. Mohammed, T.S., Rasheed, M., Al-Ani, M., Al-Shayea, Q. and Alnaimi, F. (2020), “Fault Diagnosis of Rotating Machine Based on Audio Signal Recognition System: An Efficient Approach”, International Journal of Simulation: Systems, Science & Technology, Vol. 21, no. 1, pp. 8.1-8. 
    https://doi.org/10.5013/IJSSST.a.21.01.08
  13. Müller, R.A.J., von Benda-Beckmann, A.M., Halvorsen, M.B. and Ainslie, M.A. (2020), “Application of kurtosis to underwater sound”, Acoust. Soc. Am., Vol. 148, no. 2, pp. 780- 792.
    https://doi.org/10.1121/10.0001631
  14. Wang, H. and Chen, P. (2009), “Fault Diagnosis Method Based on Kurtosis Wave and Information Divergence for Rolling Element Bearings”, WSEAS Transactions on Systems, Vol. 8, Issue 10, pp. 1155-
  15. Lemeshko, B.Yu. (2018), Kriterii proverki otkloneniya raspredeleniya ot normal'nogo zakona. Rukovodstvo po primeneniyu [Criteria for checking the deviation of the distribution from the normal law. Application guide], INFRA-M, Moscow, Russia.
  16. Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994), Continuous Univariate Distributions. Volume 1, Second Edition, John Wiley & Sons, New York, USA.
  17. Hildebrand, D.K. (1971), “Kurtosis measures bimodality?”, statist., Vol. 25, no. 1, pp. 42-43.
    https://doi.org/10.1080/00031305.1971.10477241
  18. Joiner, B.L. and Rosenblatt, J.R. (1971), “Some properties of the range in samples from Tukey's symmetric lambda distributions”, Amer. Statist. Assoc., Vol. 66, no. 334, pp. 394-399.
    https://doi.org/10.1080/01621459.1971.10482275
  19. Johnson, M.E., Tietjen, G.L. and Beckman, R.J. (1980), “A New Family of Probability Distributions with Applications to Monte Carlo Studies”, Amer. Statist. Assoc., Vol. 75, no. 370, pp. 276-279.
    https://doi.org/10.1080/01621459.1980.10477464
  20. Krasil’nikov, A.I. (2013), “Class non-Gaussian distributions with zero skewness and kurtosis”, Radioelectronics and Communications Systems, Vol. 56, no. 6, pp. 312-
    https://doi.org/10.3103/S0735272713060071
  21. Kale, B.K. and Sebastian, G. (1996), “On a Class of Symmetric Nonnormal Distributions with a Kurtosis of Three”, H.N. Nagaraja et al. (eds.). Statistical Theory and Applications. Springer-Verlag New York, Inc., pp. 55-
    https://doi.org/10.1007/978-1-4612-3990-1_6
  22. Krasilnikov, A.I. (2017), “Class of non-gaussian symmetric distributions with zero coefficient of kurtosis”, Elektronnoe modelirovanie, Vol. 39, no. 1, pp. 3-
    https://doi.org/10.15407/emodel.39.01.003
  23. Barakat, H.M. (2015), “A new method for adding two parameters to a family of distributions with application to the normal and exponential families”, Statistical Methods & Applications, Vol. 24, Issue 3, pp. 359- 
    https://doi.org/10.1007/s10260-014-0265-8
  24. Krasilnikov, A.I. (2020), “Analysis of cumulant coefficients of two-component mixtures of shifted gaussian distributions with equal variances”, Elektronnoe modelirovanie, Vol. 42, 3, pp. 71-88. 
    https://doi.org/10.15407/emodel.42.03.071
  25. Krasylnikov, O.I. (2021), “Analysis of cumulant coefficients of two-component mixtures of shifted non-Gaussian distributions”, Elektronne modeliuvannia, Vol. 43, no. 5, pp. 73- 
    https://doi.org/10.15407/emodel.43.05.073
  26. Barakat, H.M., Aboutahoun, A.W. and El-kadar, N.N. (2019), “A New Extended Mixture Skew Normal Distribution, With Applications”, Revista Colombiana de Estadstica, Vol. 42, Issue 2, pp. 167- 
    https://doi.org/10.15446/rce.v42n2.70087
  27. Vadzinskii, R.N. (2001), Spravochnik po veroiatnostnym raspredeleniiam [Directory on Probabilistic Distributions], Nauka, St. Petersburg, Russia.
  28. Krasilnikov, A.I. (2019), “Family of Subbotin distributions and its classification”, Elektronnoe modelirovanie, Vol. 41, no. 3, pp. 15-31. 
    https://doi.org/10.15407/emodel.41.03.015

Full text: PDF