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Mittag - Leffler function distribution - a new generalization of hyper-Poisson distribution

Author

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  • Subrata Chakraborty

    (Dibrugarh University)

  • S. H. Ong

    (University of Malaya)

Abstract

In this paper a new generalization of the hyper-Poisson distribution is proposed using the Mittag-Leffler function. The hyper-Poisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This Mittag-Leffler function distribution (MLFD) belongs to the generalized hypergeometric and generalized power series families and also arises as weighted Poisson distributions. MLFD is a flexible distribution with varying shapes and has a unique mode at zero or it is unimodal with one/two non-zero modes. It can be under-, equi- or over- dispersed. Various distributional properties like recurrence relation for probability mass function, cumulative distribution function, generating functions, formulas for different type of moments, their recurrence relations, index of dispersion and its classification, log-concavity, reliability properties like survival, increasing failure rate, unimodality, and stochastic ordering with respect to hyper-Poisson distribution are discussed. A particular case of the distribution is shown to arise as the steady state probability of a queuing system under state dependent service rate. The distribution has been found to fare well when compared with the hyper-Poisson and COM-Poisson type negative binomial distributions in its suitability in empirical modeling of differently dispersed count data. It is therefore expected that the proposed MLFD with its interesting features and flexibility will be a useful addition as a model for count data.

Suggested Citation

  • Subrata Chakraborty & S. H. Ong, 2017. "Mittag - Leffler function distribution - a new generalization of hyper-Poisson distribution," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-17, December.
    • Handle: RePEc:spr:jstada:v:4:y:2017:i:1:d:10.1186_s40488-017-0060-9
    DOI: 10.1186/s40488-017-0060-9
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    References listed on IDEAS

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    1. Pillai, R. N. & Jayakumar, K., 1995. "Discrete Mittag-Leffler distributions," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 271-274, May.
    2. Galit Shmueli & Thomas P. Minka & Joseph B. Kadane & Sharad Borle & Peter Boatwright, 2005. "A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(1), pages 127-142, January.
    3. repec:bot:journl:v:71:y:2011:i:4:p:501-514 is not listed on IDEAS
    4. Koichi Miyasawa, 1964. "Contributions to order statistics, edited by Ahmed E. Sarhan and Bernard G. Greenberg (John Wiley and Sons, Inc., New York, 1962), 482 pp," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 11(2), pages 227-228, June.
    5. C. Satheesh Kumar & B. Unnikrishnan Nair B. Unnikrishnan Nair, 2012. "An alternative hyper-Poisson distribution," Statistica, Department of Statistics, University of Bologna, vol. 72(3), pages 357-369.
    6. van Ophem, Hans, 2000. "Modeling Selectivity in Count-Data Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 18(4), pages 503-511, October.
    7. R. Pillai, 1990. "On Mittag-Leffler functions and related distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(1), pages 157-161, March.
    8. S. Chakraborty & S. H. Ong, 2016. "A COM-Poisson-type generalization of the negative binomial distribution," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(14), pages 4117-4135, July.
    9. Cameron, A Colin & Johansson, Per, 1997. "Count Data Regression Using Series Expansions: With Applications," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 12(3), pages 203-223, May-June.
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