ANALYSIS OF CUMULANT COEFFICIENTS OF TWO-COMPONENT MIXTURES OF SHIFTED GAUSSIAN DISTRIBUTIONS WITH EQUAL VARIANCES

A.I. Krasilnikov

Èlektron. model. 2020, 42(3):71-87
https://doi.org/10.15407/emodel.42.03.071

ABSTRACT

The dependence of cumulant coefficients of mixtures, the probability density of which can be either single-vertex or two-vertex, on the shear parameter and weight coefficients is analyzed. The ranges of possible values of cumulant coefficients are determined and the values of weighting coefficients at which cumulant coefficients are equal to zero are obtained. It is shown that the excess coefficient is zero for two values of the weight coefficients and any values of the shift parameter.

KEYWORDS

two-component mixtures of distributions, two-component Gaussian mixture, cumulant analysis, cumulant coefficients, kurtosis coefficient.

REFERENCES

  1. Aivazian, S.A., Bezhaeva, Z.I. and Staroverov, O.V. (1974) Klassifikatsiia mnogomernykh nabliudenii [Classification of multidimensional observations], Statistika, Moscow, Russia.
  2. Titterington, D.M., Smith, A.F.M. and Makov, U.E. (1985), Statistical analysis of finite mixture distributions, John Wiley & Sons, New York.
  3. McLachlan, G. and Peel, D. (2000), Finite mixture models, John Wiley & Sons, New York.
    https://doi.org/10.1002/0471721182
  4. Korolev, V.Iu. (2008), Veroiatnostno-statisticheskii analiz khaoticheskikh protsessov s pomoshchiu smeshannykh gaussovskikh modelei. Dekompozitsiia volatilnosti finansovykh indeksov i turbulentnoi plazmy [Probabilistic-statistical analysis of chaotic processes using mixed Gaussian models. Decomposition of volatility of financial indices and turbulent plasma], Izd-vo Instituta problem informatiki RAN, Moscow, Russia.
  5. Aprausheva, N.N. and Sorokin, S.V. (2015), Zametki o gaussovykh smesiakh [Notes on Gaussian mixtures], VTs Rossiiskoi akademii nauk, Moscow, Russia.
  6. Punzo, A. and McNicholas, P.D. (2016), “Parsimonious mixtures of multivariate contaminated normal distributions”, Preprint submitted to arXiv 1305.4669, May 20, 2016. – pp. 1–28, available at: https://arxiv.org/pdf/1305.4669.pdf
  7. Chabdarov, Sh.M., Safiullin, N.Z. and Feoktistov A.Iu. (1983), Osnovy statisticheskoi teorii radiosviazi. Poligaussovy modeli i metody [Fundamentals of the statistical theory of radio communications. Poly-Gaussian models and methods], Kazanskii aviats. in-t im. Tupoleva, Kazan', Russia.
  8. Sorokin, V.N., V'iugin, V.V. and Tananykin, A.A. (2012), “Voice Recognition: An Analytical Review”, Informatsionnye protsessy, Vol. 12, no. 1, pp. 1–30.
  9. Tukey, J.W. (1960), “A survey of sampling from contaminated distributions”, in Olkin, I. (Ed.), Contributions to Probability and Statistics, Stanford Univ. Press, Stanford, pp. 448–485.
  10. Brashevan, A.N. (2004) “A statistical model of multimode experimental data”, Radioelektronni i kompiuterni systemy, no. 1 (5), pp. 105–108.
  11. Litvak, M.Ia. and Maliugin, V.I. (2012), “Poly-Gaussian models of a non-Gaussian randomly roughened surface”, Zhurnal tekhnicheskoi fiziki, Vol. 82, no. 4, pp. 99–107.
    https://doi.org/10.1134/S1063784212040172
  12. Rubtsov, E.A. (2014), “Distribution of errors of aircraft coordinates determination”, Vestnik Samarskogo gos. aerokosmicheskogo universiteta, no. 1 (43), pp. 267–274.
    https://doi.org/10.18287/1998-6629-2014-0-1(43)-267-274
  13. Krasilnikov, A.I. and Pilipenko, K.P. (2007), “Unimodal two-componental Gaussian mixture. Excess kurtosis”, Elektronika i sviaz, no. 2 (37), pp. 32–38.
  14. Krasilnikov,I. and Pilipenko, K.P. (2008), “Application of a two-component Gaussian mixture to identify single-peak symmetric probability density functions”, Elektronika i sviaz, no. 5 (46), pp. 20–29.
  15. Chepynoha,V. (2010), “Areas of realization of bi-Gaussian models of skewness-excess random variables with the punched moment-cumulant description”, Visnyk ChDTU, no. 2, pp. 91–95.
  16. Tokmachev, M.S. and Smirnov, S.V. (2012), “Software implementation of the study of mixtures of probability distributions”, Vestnik Novgorodskogo gos. universiteta, no. 68, pp. 85–89.
  17. Kunchenko,P., Zabolotnii, S.V., Koval, V.V. and Chepynoha, A.V. (2005), “Simulation of excess random variables with given cumulative description based on bi-Gaussian distribution”, Visnyk ChDTU, no. 1, pp. 38–42.
  18. Krasilnikov, A.I. (2018), “Modeling of perforated random variables on the basis of mixtures of shifted distributions”, Elektronnoe modelirovanie, Vol. 40, no. 1, pp. 47–61.
    https://doi.org/10.15407/emodel.40.01.047

Full text: PDF