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Characterizations and Infinite Divisibility of Extended COM-Poisson Distribution

Received: 9 August 2015     Accepted: 26 August 2015     Published: 27 August 2015
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Abstract

This article provides some characterizations of extended COM-Poisson distribution: conditional distribution given the sum, functional operator characterization (Stein identity). We also give some conditions such that the extended COM-Poisson distribution is infinitely divisible, hence some subclass of extended COM-Poisson distributions are discrete compound Poisson distribution.

Published in International Journal of Statistical Distributions and Applications (Volume 1, Issue 1)
DOI 10.11648/j.ijsd.20150101.11
Page(s) 1-4
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

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Keywords

Conway-Maxwell-Poisson distribution, conditional distribution, discrete compound Poisson distribution, infinitely divisible, Stein identity

References
[1] Balabdaoui, F., Jankowski, H., Rufibach, K., Pavlides, M. (2013). Asymptotics of the discrete log-concave maximum likelihood estimator and related applications. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(4), 769-790.
[2] Barriga, G. D., Louzada, F. (2014). The zero-inated Conway-Maxwell-Poisson distribution: Bayesian inference, regression modeling and inuence diagnostic. Statistical Methodology, 21, 23-34.
[3] Brown, T. C., Xia, A. (2001). Stein's method and birth-death processes. Annals of probability, 1373-1403.
[4] Buchmann, B., Grbel, R. (2003). Decompounding: an estimation problem for Poisson random sums. The Annals of Statistics,31(4), 1054-1074.
[5] Chakraborty, S. (2015). A new extension of Conway-Maxwell-Poisson distribution and its properties. arXiv preprint arXiv:1503.04443.
[6] Chakraborty, S., Ong, S. H. (2014), A COM-type Generalization of the Negative Binomial Distribution, Accepted in April 2014, to appear in Communications in Statistics-Theory and Methods
[7] Conway, R. W., Maxwell, W. L. (1962). A queuing model with state dependent service rates. Journal of Industrial Engineering, 12(2), 132-136.
[8] Daly, F., Gaunt, R. E. (2015). The Conway-Maxwell-Poisson distribution: distributional theory and approximation. arXiv preprint arXiv:1503.07012.
[9] Feller, W. (1971). An introduction to probability theory and its applications,Vol. I. 3rd., Wiley, New York.
[10] Guikema, S. D., Goffelt, J. P. (2008). A exible count data regression model for risk analysis. Risk analysis, 28(1), 213-223.
[11] Imoto, T. (2014). A generalized Conway–Maxwell–Poisson distribution which includes the negative binomial distribution. Applied Mathematics and Computation, 247, 824-834.
[12] Kagan, A. M., Linnik, Y. V., Rao, C. R. (1973). Characterization problems in mathematical statistics, Wiley.
[13] Khan, N. M., Khan, M. H. (2010). Model for Analyzing Counts with Over-, Equi-and Under-Dispersion in Actuarial Statistics. Journal of Mathematics and Statistics, 6(2), 92-95.
[14] Kokonendji, C. C. (2014). Over-and Underdispersion Models. Methods and Applications of Statistics in Clinical Trials: Planning, Analysis, and Inferential Methods, Volume 2, 506-526.
[15] Patil, G. P., Seshadri, V. (1964). Characterization theorems for some univariate probability distributions. Journal of the Royal Statistical Society. Series B (Methodological), 286-292.
[16] Sellers, K. F., Shmueli, G. (2010). Predicting censored count data with COM-Poisson regression. Robert H. Smith School Research Paper No. RHS-06-129.
[17] Rodrigues, J., de Castro, M., Cancho, V. G., Balakrishnan, N. (2009). COM-Poisson cure rate survival models and an application to a cutaneous melanoma data. Journal of Statistical Planning and Inference, 139(10), 3605-3611.
[18] Saghir, A., Lin, Z., Abbasi, S. A., Ahmad, S. (2013). The Use of Probability Limits of COM-Poisson Charts and their Applications. Quality and Reliability Engineering International, 29(5), 759-770.
[19] Saumard, A., Wellner, J. A. (2014). Log-concavity and strong log-concavity: a review. Statistics Surveys, 8, 45-114.
[20] Sellers, K. F., Borle, S., Shmueli, G. (2012). The COM-Poisson model for count data: a survey of methods and applications. Applied Stochastic Models in Business and Industry, 28(2), 104-116.
[21] Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., Boatwright, P. (2005). A useful distribution for ftting discrete data: revival of the Conway-Maxwell-Poisson distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(1),127-142.
[22] Steutel, F. W. (1970). Preservation of infinite divisibility under mixing and related topics. MC Tracts, 33, 1-99.
[23] Steutel, F. W., Van Harn, K. (2003). Infinite divisibility of probability distributions on the real line. CRC Press, New York.
[24] Zhang, H., He J., Huang H. (2013). On nonnegative integer-valued Lévy processes and applications in probabilistic number theory and inventory policies American Journal of Theoretical and Applied Statistics, 2 (5), 110-121.
[25] Zhang, H., Liu, Y., Li, B. (2014). Notes on discrete compound Poisson model with applications to risk theory. Insurance: Mathematics and Economics, 59, 325-336.
[26] Zhu, F. (2012). Modeling time series of counts with COM-Poisson INGARCH models. Mathematical and Computer Modelling, 56(9), 191-203.
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    Huiming Zhang. (2015). Characterizations and Infinite Divisibility of Extended COM-Poisson Distribution. International Journal of Statistical Distributions and Applications, 1(1), 1-4. https://doi.org/10.11648/j.ijsd.20150101.11

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    ACS Style

    Huiming Zhang. Characterizations and Infinite Divisibility of Extended COM-Poisson Distribution. Int. J. Stat. Distrib. Appl. 2015, 1(1), 1-4. doi: 10.11648/j.ijsd.20150101.11

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    AMA Style

    Huiming Zhang. Characterizations and Infinite Divisibility of Extended COM-Poisson Distribution. Int J Stat Distrib Appl. 2015;1(1):1-4. doi: 10.11648/j.ijsd.20150101.11

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  • @article{10.11648/j.ijsd.20150101.11,
      author = {Huiming Zhang},
      title = {Characterizations and Infinite Divisibility of Extended COM-Poisson Distribution},
      journal = {International Journal of Statistical Distributions and Applications},
      volume = {1},
      number = {1},
      pages = {1-4},
      doi = {10.11648/j.ijsd.20150101.11},
      url = {https://doi.org/10.11648/j.ijsd.20150101.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijsd.20150101.11},
      abstract = {This article provides some characterizations of extended COM-Poisson distribution: conditional distribution given the sum, functional operator characterization (Stein identity). We also give some conditions such that the extended COM-Poisson distribution is infinitely divisible, hence some subclass of extended COM-Poisson distributions are discrete compound Poisson distribution.},
     year = {2015}
    }
    

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    T2  - International Journal of Statistical Distributions and Applications
    JF  - International Journal of Statistical Distributions and Applications
    JO  - International Journal of Statistical Distributions and Applications
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    PB  - Science Publishing Group
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    AB  - This article provides some characterizations of extended COM-Poisson distribution: conditional distribution given the sum, functional operator characterization (Stein identity). We also give some conditions such that the extended COM-Poisson distribution is infinitely divisible, hence some subclass of extended COM-Poisson distributions are discrete compound Poisson distribution.
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Author Information
  • School of Mathematics and Statistics, Central China Normal University, Wuhan, China

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