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ON THE DISCRETE DISTRIBUTION GENERATED BY LEVY PROBABILITY

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Abstract

In this paper, we consider one-parameter regularly varying discrete distribution generated by Levy probability (RDLP). Some useful plots of the models are illustrated. Mathematically, to propose the RDLP model as a discrete distribution in bioinformatics, some common statistical facts such as unimodality, skewness to the right, upward/downward convexity, regular variation at infinity, and asymptotically constant slowly varying component are established for the model. Based on the Monte Carlo method and limited-memory Broyden–Fletcher–Goldfarb–Shanno for bound-constrained optimization, simulation studies are done to estimate bias, mean square error, and maximum likelihood estimation for the unknown parameter of the model. Some asymptotic properties and asymptotic expansions are given for useful functions. Moreover, we shall attempt to examine the RDLP model with a real count data set to show its application, and also compare it with two rival models.

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Data availability

The authors confirm that the data supporting the findings of this study are available within the article.

Code Availability

Simulation studies, drawing figures and fitting the models have been done with the help of R statistical software. The R codes are available from the corresponding author upon request.

References

  1. J. Astola, E. Danielian, Frequency Distributions in Biomolecular Systems and Growing Networks, Tampere International Center for Signal Processing (TICSP), Series no. 31, Tampere, Finland, (2007).

  2. J. Astola, E. Danielian, S. Arzumanyan, Frequency distributions in bioinformatics, a review, Proceedings of the Yerevan State University: Physical Mathematical Sciences, 223(3) (2010), 3-22.

    Google Scholar 

  3. D. Bhati, H. S. Bakouch, A new infinitely divisible discrete distribution with applications to count data modeling, Communications in Statistics - Theory and Methods, 48(6) (2019), 1401-1416.

    Article  MathSciNet  Google Scholar 

  4. A. A. Borovkov, Mathematical Statistics, Gordon and Breach Science Publishers, (1998), (translated from the Russian by A. Moullagaliev).

  5. R. H. Byrd, P. Lu, J. Nocedal, C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM Journal of Scientific Computing, 16(5) (1995), 1190-1208.

    Article  MathSciNet  Google Scholar 

  6. E. Danielian, J. Astola, On regularly varying hypergeometric distributions, Proceedings International TICSP Workshop on Spectral Methods and Multirate Signal Processing, Florence, Italy, 2-3 September; In Astola et al. (eds.): TICSP Series 34, Tampere, Finland, pp. 127-132, (2006).

  7. E. Danielian, R. Chitchyan, D. Farbod, On a new regularly varying generalized hypergeometric distribution of the second type, Mathematical Reports, 18(68)(2) (2016), 217-232.

  8. D. Farbod, The asymptotic properties of some discrete distributions generated by Levy’s law, Far East Journal of Theoretical Statistics, 26(1) (2008), 121-128.

    MathSciNet  Google Scholar 

  9. D. Farbod, M-estimators as GMM for Stable Laws Discretizations, Journal of Statistical Research of Iran, 8(1) (2011), 85-96.

    Article  Google Scholar 

  10. D. Farbod, Some statistical inferences for two frequency distributions arising in bioinformatics, Applied Mathematics E-Notes, 14 (2014), 151-160.

    MathSciNet  Google Scholar 

  11. D. Farbod, Modeling and simulation studies for some truncated discrete distributions generated by stable densities, Mathematical Sciences, Springer, 16(2) (2022), 105-114.

    Article  MathSciNet  Google Scholar 

  12. D. Farbod, K. Arzideh, Asymptotic properties of moment estimators for distributions generated by Levys law, International Journal of Applied Mathematics and Statistics, 20(M11) (2011), 55-59.

  13. D. Farbod, K. Arzideh, On the properties of a parametric function for distribution generated by Levy law, International Journal of Mathematics and Computation, 20(3) (2013), 52-59.

    MathSciNet  Google Scholar 

  14. D. Farbod, K. Gasparian, Asymptotic properties of maximum likelihood estimator for some discrete distributions generated by Cauchy stable law, STATISTICA, 68(3-4), (2008), 321-326.

    Google Scholar 

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

    Google Scholar 

  16. G. H. Given, J. A. Hoeting, Computational Statistics, Wiley and Sons, Second edition, (2012).

  17. I. S. Gradshteyn, I. M. Ryznik, Tables of Integrals, Series and Products, Academic Press, translated from Russian by S. Technica, edited by A. Jeffrey and D. Zwillinger, (2007).

  18. J. Keilson, H. Gerber, Some results for discrete unimodality, Journal of the American Statistical Association, 66 (1971), 386-389.

    Article  Google Scholar 

  19. V. A. Kuznetsov, Distributions associated with stochastic processes of gene expression in a single eukaryotic cell, EURASIP Journal on Applied Signal Processing, 4 (2001) 258-296.

    Google Scholar 

  20. V. A. Kuznetsov, Mathematical modeling of avidity distribution and estimating general binding properties of transcription factors from genome-wide binding profiles, In: Tatarinova,T. V. and Nikolsky, Y. (eds), Biological Networks and Pathway Analysis, pp. 193-276, Springer, New York, (2017).

    Chapter  Google Scholar 

  21. V. A. Kuznetsov, A. Grageda, D. Farbod, Generalized hypergeometric distributions generated by birth-death process in bioinformatics, Markov Processes and Related Fields, 28(2) (2022), 303-327.

    MathSciNet  Google Scholar 

  22. J. P. Nolan, Basic Properties of Univariate Stable Distributions. In: Univariate Stable Distributions. Springer Series in Operations Research and Financial Engineering. Springer, (2020).

  23. C. Zhu, R. H. Byrd, P. Lu, J. Nocedal, L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization, ACM Transactions on Mathematical Software, 23(4) (1997), 550-560.

    Article  MathSciNet  Google Scholar 

  24. J. Yang, Protein secondary structure prediction based on neural network models and support vector machines, CS229 Final Project, December, (2008).

  25. V. M. Zolotarev, One-dimensional Stable Distributions, Vol. 65 of Translation of Mathematical Monographs, American Mathematical Society, (1986) (translated from the original 1983 Russian edition).

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Acknowledgements

The authors would like to thank the Editor-in-Chief, Associate Editor and the three anonymous referees for their valuable and constructive suggestions which led to significant improvements of the quality of our paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Engineering Science, Quchan University of Technology, Quchan, Iran

    Davood Farbod

  2. Department of Mathematics and Statistics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

    Maryam Basirat

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  1. Davood Farbod
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  2. Maryam Basirat
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Correspondence to Davood Farbod.

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Farbod, D., Basirat, M. ON THE DISCRETE DISTRIBUTION GENERATED BY LEVY PROBABILITY. J Math Sci 280, 198–211 (2024). https://doi.org/10.1007/s10958-023-06808-0

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