Modeling and simulation studies for some truncated discrete distributions generated by stable densities


Some discrete distributions generated by stable densities (DGSDs) could be considered as models for describing phenomena arising in bioinformatics. Since probability mass and distribution functions are not in closed forms, simulation studies and real applications of these distributions have not been studied yet. To do this, we need to consider DGSDs as truncated, namely, truncated DGSDs (T-DGSDs). In this paper, some statistical properties of the T-DGSDs models are established. Based on the Monte Carlo method, limited-memory Broyden–Fletcher–Goldfarb–Shanno for bound-constrained optimization, and Nelder–Mead optimization algorithms, we do a simulation to estimate biases, mean square errors, and maximum likelihood estimations for the unknown parameters of the T-DGSDs. Moreover, we fit these T-DGSDs models with some real data sets in bioinformatics and then compare them to some frequency distributions.

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  1. 1.

    Astola, J., Danielian, E.: Frequency distributions in biomolecular systems and growing networks. Tampere International Center for Signal Processing, Series no. 31, Tampere, Finland, (2007)

  2. 2.

    Astola, J., Danielian, E., Arzumanyan, S.: Frequency distributions in bioinformatics, a review. Proc. Yerevan State Univ.: Phys. Math. Sci. 223(3), 3–22, (2010)

  3. 3.

    Borovkov, A.A.: Mathematical statistics. Gordon and Breach Science Publishers (1998)

  4. 4.

    Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comp. 16(5), 1190–1208 (1995)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Farbod, D.: The asymptotic properties of some discrete distributions generated by symmetric stable laws. Inform. Tech. Manag. 10, 49–55 (2007)

    Google Scholar 

  6. 6.

    Farbod, D.: M-estimators as GMM for Stable laws discretizations. J. Statist. Res. Iran. 8(1), 85–96 (2011)

    Article  Google Scholar 

  7. 7.

    Farbod, D.: M-estimation of the tail index for discrete distribution generated by stable density. Presented at the \(14{\rm th}\) Iranian statistics conference, Shahrood University of Technology, Shahrood, Iran, (2018)

  8. 8.

    Farbod, D., Gasparian, K.: The asymptotic properties of discrete distributions generated by standard skewed stable laws. Proceedings \(9^{th}\) Iranian Statistics Conference, University of Isfahan, Isfahan, Iran, 138–144, (2008)

  9. 9.

    Farbod, D., Gasparian, K.: The asymptotic properties of maximum likelihood estimators for some discrete distributions generated by Cauchy stable law. Statistica 68(3–4), 321–326 (2008)

    MATH  Google Scholar 

  10. 10.

    Farbod, D., Gasparian, K.: On the confidence intervals of parametric functions for distributions generated by symmetric stable laws. Statistica 72(4), 405–413 (2012)

    MATH  Google Scholar 

  11. 11.

    Farbod, D., Gasparian, K.: On the maximum likelihood estimators for some generalized Pareto-like frequency distribution. J. Iran. Stat. Soc. (JIRSS) 12(2), 211–233 (2013)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Given, G.H., Hoeting, J.A.: Computational statistics, 2nd edn. Wiley (2012)

  13. 13.

    Ivchenko, G.I., Medvedev, Yu.I.: Mathematical statistics. Mir Press, Moscow (1990)

  14. 14.

    Kuznetsov, V.A.: Distribution associated with stochastic processes of gene expression in a single eukaryotic cell. EURASIP J. Adv. Signal Proc. 1, 285–296 (2001)

    Article  Google Scholar 

  15. 15.

    Kuznetsov, V.A.: Stochastic model of evolution of conserved protein coding sequences. AIP conference proceedings, American Institute of Physics 665, 369–380 (2003)

  16. 16.

    Kuznetsov, V.A., Pickalov, V.A., Senko, O.V., Knott, G.D.: Analysis of the evolving proteomes: predictions of the number for protein domains in nature and the number of genes in eukaryotic organisms. J. Biol. Syst. 10(4), 381–407 (2002)

    Article  Google Scholar 

  17. 17.

    Lehmann, E.L.: Theory of Point Estimation. Wiley (1983)

  18. 18.

    Loh, Y., Wu, Q., Chew, J., Vega, V., Zhang, W., Chen, X., Bourque, G., George, J., Leong, B., Liu, J., Wong, K., Sung, K., Lee, Ch., Zhao, X., Chiu, K., Lipovich, L., Kuznetsov, V., Robson, P., Stanton, L., Wei, Ch., Ruan, Y., Lim, B., Ng, H.: The Oct4 and Nanog transcription network regulates pluripotency in mouse embryonic stem cell. Nat. Genet. 38(4), 431–440 (2006)

    Article  Google Scholar 

  19. 19.

    Matsui, M., Takemura, A.: Some improvements in numerical evaluation of symmetric stable density and its derivatives. Comm. Stat. Th. Meth. 35(1), 149–172 (2006)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308–313 (1965)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Nolan, J.P.: Stable distributions: models for heavy-tailed data. Birkhauser (2010)

  22. 22.

    Shiryaev, A.N.: Probability Theory (Graduate Text in Mathematics). Springer (1996)

  23. 23.

    Wongsurawat, T., Jenjaroenpun, P., Kwoh, C.K., Kuznetsov, V.: Quantitative model of R-loop forming structures reveals a novel level of RNA-DNA interactome complexity. Nucl. Acids Res. 40(2), e16 (2012)

    Article  Google Scholar 

  24. 24.

    Zhu, C., Byrd, R.H., Lu, P., Nocedal, J.: L-BFGS-B: algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization. ACM Trans. Math. Soft. 23(4), 550–560 (1997)

    Article  Google Scholar 

  25. 25.

    Zolotarev, V. M.: One-dimensional stable distributions. American Mathematical Society (translated from original 1983 Russian edition), (1986)

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The author would like to thank the two anonymous referees for their valuable suggestions and comments. This work was supported by Quchan University of Technology under a grant number 7226.

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Correspondence to Davood Farbod.

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This work was supported by Quchan University of Technology under a grant number 7226.

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Farbod, D. Modeling and simulation studies for some truncated discrete distributions generated by stable densities. Math Sci (2021).

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  • Discrete distributions generated by stable densities
  • Maximum likelihood estimation
  • Monte Carlo method

Mathematics Subject Classification

  • 62F10
  • 60E07
  • 62P10