COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data

Abstract

We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein’s identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.

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References

  1. 1.

    Arnold T B, Emerson J W. Nonparametric goodness-of-fit tests for discrete null distributions. The R Journal, 2011, 3(2): 34–39

    Google Scholar 

  2. 2.

    Böhning D. Ratio plot and ratio regression with applications to social and medical sciences. Statist Sci, 2016, 31(2): 205–218

    MathSciNet  Article  Google Scholar 

  3. 3.

    Borges P, Rodrigues J, Balakrishnan N, Bazán J. A COM-Poisson type generalization of the binomial distribution and its properties and applications. Stat Probab Lett, 2014, 87: 158–166

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Brown T C, Phillips M J. Negative binomial approximation with Stein’s method. Methodol Comput Appl Probab, 1999, 1(4): 407–421

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Brown T C, Xia A. Stein’s method and birth-death processes. Ann Probab, 2001, 29(3): 1373–1403

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Chakraborty S, Ong S H. A COM-Poisson-type generalization of the negative binomial distribution. Comm Statist Theory Methods, 2016, 45(14): 4117–4135

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Chakraborty S, Imoto T. Extended Conway-Maxwell-Poisson distribution and its properties and applications. J Stat Distrib App, 2016, 3: 5

    Article  MATH  Google Scholar 

  8. 8.

    Conway R W, Maxwell W L. A queuing model with state dependent service rates. J Industrial Engineering, 1962, 12(2): 132–136

    Google Scholar 

  9. 9.

    Denuit M, Maréchal X, Pitrebois S, Walhin J-F. Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems. Chichester: John Wiley & Sons, 2007

    Google Scholar 

  10. 10.

    Gómez-Déniz E, Sarabia José María, Calderín-Ojeda E. A new discrete distribution with actuarial applications. Insurance Math Econom, 2011, 48(3): 406–412

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Gómez-Déniz E, Calderín-Ojeda E. Unconditional distributions obtained from conditional specification models with applications in risk theory. Scand Actuar J, 2014, (7): 602–619

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Gupta R C, Sim S Z, Ong S H. Analysis of discrete data by Conway-Maxwell Poisson distribution. Adv Stat Anal, 2014, 98(4): 327–343

    MathSciNet  Article  Google Scholar 

  13. 13.

    Haberman S J. A warning on the use of chi-squared statistics with frequency tables with small expected cell counts. J Amer Statist Assoc, 1988, 83(402): 555–560

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Ibragimov I A. On the composition of unimodal distributions. Theory Probab Appl, 1956, 1(2): 255–260

    MathSciNet  Article  Google Scholar 

  15. 15.

    Imoto T. A generalized Conway-Maxwell-Poisson distribution which includes the negative binomial distribution. Appl Math Comput, 2014, 247: 824–834

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Johnson N L, Kemp A W, Kotz S. Univariate Discrete Distributions. 3rd ed. Hoboken: John Wiley & Sons, 2005

    Google Scholar 

  17. 17.

    Kadane J B. Sums of possibly associated Bernoulli variables: the Conway-Maxwell-Binomial distribution. Bayesian Anal, 2016, 11(2): 403–420

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Kagan A M, Rao C R, Linnik Y V. Characterization Problems in Mathematical Statistics. New York: Wiley

  19. 19.

    Kaas R, Goovaerts M, Dhaene J, Denuit M. Modern Actuarial Risk Theory Using R. 2nd ed. Berlin; Springer, 2008

    Google Scholar 

  20. 20.

    Kokonendji C C, Mizere D, Balakrishnan N. Connections of the Poisson weight function to overdispersion and underdispersion. J Statist Plann Inference, 2008, 138(5): 1287–1296

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Liu Q, Lee J, Jordan M. A kernelized Stein discrepancy for goodness-of-fit tests. International Conference on Machine Learning, 2016, 276–284

    Google Scholar 

  22. 22.

    Meintanis S G, Nikitin Y Y. A class of count models and a new consistent test for the Poisson distribution. J Statist Plann Inference, 2008, 138(12): 3722–3732

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Patil G P, Seshadri V. Characterization theorems for some univariate probability distributions. J R Stat Soc Ser B Stat Methodol, 1964, 26: 286–292

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Rao C R, Rubin H. On a characterization of the Poisson distribution. Sankhyā, 1964, 32(2–3): 295–298

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Rényi A. On measures of entropy and information. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol 1. Contributions to the Theory of Statistics. The Regents of the University of California, 1961, 547–561

    Google Scholar 

  26. 26.

    Ramalingam S, Jagbir S. A characterization of the logarithmic series distribution and its application. Comm Statist Theory Methods, 1984, 13(7): 865–875

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Rodrigues J, de Castro M, Cancho V G, Balakrishnan N. COM-Poisson cure rate survival models and an application to a cutaneous melanoma data. J Statist Plann Inference, 2009, 139(10): 3605–3611

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Sellers K F, Borle S, Shmueli G. The COM-Poisson model for count data: a survey of methods and applications. Appl Stoch Models Bus Ind, 2012, 28(2): 104–116

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Shaked M, Shanthikumar J G. Stochastic Orders. Berlin: Springer, 2007

    Google Scholar 

  30. 30.

    Shanbhag D N. An extension of the Rao-Rubin characterization of the Poisson distribution. J Appl Probab, 1977, 14(3): 640–646

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Shmueli G, Minka T P, Kadane J B, Borle S, Boatwright P. A useful distribution for fitting discrete data: revival of the Conway-Maxwell-Poisson distribution. J R Stat Soc Ser C Appl Stat, 2005, 54(1): 127–142

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Steutel F W. Preservation of Infinite Divisibility under Mixing and Related Topics. Math Centre Tracts, 33. Amsterdam: Mathematisch Centrum, 1970

    Google Scholar 

  33. 33.

    Steutel F W, van Harn K. Infinite Divisibility of Probability Distributions on the Real Line. Boca Raton: CRC Press, 2003

    Google Scholar 

  34. 34.

    Tsallis C. Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys, 1988, 52(1–2): 479–487

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Temme N M. Special Functions: An Introduction to the Classical Functions of Mathematical Physics. New York: John Wiley & Sons, 2011

    Google Scholar 

  36. 36.

    Wang L M, Lei Y L. Simulation and EM algorithm for the distribution of number of claim in the heterogeneous portfolio. Commun Appl Math Comput Sci, 2000, 14(2): 71–78

    Google Scholar 

  37. 37.

    Willmot G E. The Poisson-inverse Gaussian distribution as an alternative to the negative binomial. Scand Actuar J, 1987, (3–4): 113–127

    MathSciNet  Article  Google Scholar 

  38. 38.

    Wimmer G, Köhler R, Grotjahn R, Altmann G. Towards a theory of word length distribution. J Quant Linguist, 1994, 1(1): 98–106

    Article  Google Scholar 

  39. 39.

    Wimmer G, Altmann G. Thesaurus of Univariate Discrete Probability Distributions. Essen: Stamm, 1999

    Google Scholar 

  40. 40.

    Zhang H, Liu Y, Li B. Notes on discrete compound Poisson model with applications to risk theory. Insurance Math Econom, 2014, 59: 325–336

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Zhang H, Li B. Characterizations of discrete compound Poisson distributions. Comm Statist Theory Methods, 2016, 45(22): 6789–6802

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Zhang H, Li B, Kerns G J. A characterization of signed discrete infinitely divisible distributions. Studia Sci Math Hung, 2017, 54(4): 446–470

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Zhang H, Jia J. Elastic-net regularized high-dimensional negative binomial regression: consistency and weak signals detection. arXiv: 1712.03412

  44. 44.

    Zygmund A. Trigonometric Series. Cambridge: Cambridge University Press, 2002

    Google Scholar 

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Acknowledgements

The proposed COM-negative binomial distribution of this work was as early as conceptualized in December, 2014 when the authors saw the online version of [15]. The authors want to thank Prof. R. Köhler for mailing the valuable encyclopedia of discrete univariate distributions [39] to them. This work was partly supported by the National Natural Science Foundation of China (Grant No. 11201165).

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Correspondence to Huiming Zhang.

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Zhang, H., Tan, K. & Li, B. COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data. Front. Math. China 13, 967–998 (2018). https://doi.org/10.1007/s11464-018-0714-z

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Keywords

  • Overdispersion
  • zero-inflated data
  • infinite divisibility
  • Stein’s characterization
  • discrete Kolmogorov-Smirnov test

MSC

  • 60E07
  • 60A05
  • 62F10