RESEARCH OF APPROXIMATION ACCURACY OF SYMMETRICAL PROBABILITY DENSITY FUNCTION BY ORTHOGONAL REPRESENTATIONS USING HERMITE POLYNOMIALS

V.S. Beregun

Èlektron. model. 2018, 38(3):87-98
https://doi.org/10.15407/emodel.38.03.087

ABSTRACT

Approximation errors of symmetrical probability density functions of typical distributions using the Hermite polynomials have been described. Comparison of results for the partial and generalized sums of series for the Hermite polynomials is carried out at the number of series components to twenty. Orthogonal representations have been checked for non-negativity.

KEYWORDS

probability density function, approximation, Hermite polynomials, orthogonal representation, generalized summation.

REFERENCES

1. Shelukhin, O.I. (1998), Negaussovskie protsessy v radiotekhnike [Non-gaussian processes in radio engineering], Radio i sviaz, Moscow, Russia.
2. Novitskii, P.V. and Zograf, I.A. (1991), Otsenka pogreshnostei rezultatov izmerenii [Error estimation in measurement results], Energoatomizdat, St. Petersburg, Russia.
3. Dech, R. (1965), Nelineinye preobrazovaniia sluchainykh protsessov [Nonlinear trasformations of random processes], Translated by Smirenin, B.A., Sov. radio, Moscow, Russia.
4. Krasilnikov, A.I. (2014), Modeli shumovykh signalov v sistemakh diagnostiki teploenergeticheskogo oborudovaniia [Models of noise signals in systems of diagnostics of the heatand-
power equipment], Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine.
5. Kramer, G. (1975), Matematicheskie metody statistiki [Mathematical methods of statistics], Translated by Monin, A.S. and Petrov, A.A., Mir, Moscow, Russia.
6. Kendall, M. and Stiuart, A. (1966), Teoriia raspredelenii [Distribution theory], Translated by Sazonov, V.V. and Shiriaev, A.N., Nauka, Moscow, Russia.
7. Devroi, L. and Derfi, L. (1988), Neparametricheskoe otsenivanie plotnosti. L1-podkhod [Nonparametric density estimation. The L1 view], Translated by Tsybakov, A.B., Mir, Moscow, Russia.
8. Krasylnikov, O.I. and Beregun, V.S. (2010), “Ordering of orthogonal representations of probability density functions of stochastic processes”, Elektronika ta systemy upravlinnia, no. 3 (25), pp. 28-35.
9. Suetin, P.K. (2005), Klassicheskie ortogonalnye polinomy [Classical orthogonal polynoms], Fizmatlit, Moscow, Russia.
10. Senatov, V.V. (2009), Tsentralnaia predelnaia teorema: Tochnost approksimatsii i asimptoticheskie razlozheniia [Central limit theorem: approximation accuracy and asymptotic decompositions], Knizhnyi dom “Librokom”, Moscow, Russia.
11. Krasilnikov, A.I. and Beregun, V.S. (2015), “Using of the generalized summation methods at approximation of probability density function”, Elektronnoe modelirovanie, Vol. 37, no. 2, pp. 87-99.
12. Vadzinskii, R.N. (2001), Spravochnik po veroiatnostnym raspredeleniiam [Directory on probabilistic distributions], Nauka, St. Petersburg, Russia.
13. Malakhov, A.N. (1978), Kumuliantnyi analiz sluchainykh negaussovykh protsessov i ikh preobrazovanii [Cumulant analysis of random non-gaussian processes and their transformations], Sov. radio, Moscow, Russia.
14. Jondeau, E. and Rockinger, M. (2001), “Gram-Charlier densities”, Journal of Economic Dynamics & Control, Vol. 25, pp. 1457-1483.
https://doi.org/10.1016/S0165-1889(99)00082-2
15. Zapevalov, A.S. and Ratner, B.Yu. (2003), “Analytical model of probability density function of sea surface slopes”, Morskoi gidrofizicheskii zhurnal, no. 1, pp. 3-17.
16. Beregun, V.S. and Krasylnikov, O.I. (2010), “Research of non-negativity areas at orthogonal representations of probability density function”, Elektronika i sviaz, no. 3 (56), pp. 73-78.
17. The Bernoulli number page, available at http://bernoulli.org/
18. The on-line encyclopedia of integer sequences, available at: http://oeis.org/A028296.

Full text: PDF (in Russian)