Frontiers of Mathematics in China >
COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inated count data
Received date: 20 Jun 2017
Accepted date: 05 Jul 2018
Published date: 14 Aug 2018
Copyright
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-inated data sets. With the aid of ratio regression, we employ maximum likeli-hood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.
Huiming ZHANG , Kai TAN , Bo LI . COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inated count data[J]. Frontiers of Mathematics in China, 2018 , 13(4) : 967 -998 . DOI: 10.1007/s11464-018-0714-z
1 |
Arnold T B, Emerson J W. Nonparametric goodness-of-fit tests for discrete null distributions. The R Journal, 2011, 3(2): 34–39
|
2 |
Böhning D. Ratio plot and ratio regression with applications to social and medical sciences. Statist Sci, 2016, 31(2): 205–218
|
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
|
4 |
Brown T C, Phillips M J. Negative binomial approximation with Stein's method. Methodol Comput Appl Probab, 1999, 1(4): 407–421
|
5 |
Brown T C, Xia A. Stein's method and birth-death processes. Ann Probab, 2001, 29(3): 1373–1403
|
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
|
7 |
Chakraborty S, Imoto T. Extended Conway-Maxwell-Poisson distribution and its properties and applications. J Stat Distrib App, 2016, 3: 5
|
8 |
Conway R W, Maxwell W L. A queuing model with state dependent service rates. J Industrial Engineering, 1962, 12(2): 132–136
|
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
|
10 |
Gómez-Déniz E, Sarabia José María E, Calderín-Ojeda
|
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
|
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
|
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
|
14 |
Ibragimov I A. On the composition of unimodal distributions. Theory Probab Appl, 1956, 1(2): 255–260
|
15 |
Imoto T. A generalized Conway-Maxwell-Poisson distribution which includes the negative binomial distribution. Appl Math Comput, 2014, 247: 824–834
|
16 |
Johnson N L, Kemp A W, Kotz S. Univariate Discrete Distributions.3rd ed.Hoboken: John Wiley & Sons, 2005
|
17 |
Kadane J B. Sums of possibly associated Bernoulli variables: the Conway-Maxwell-Binomial distribution. Bayesian Anal, 2016, 11(2): 403–420
|
18 |
Kagan A M, Rao C R, Linnik Y V. Characterization Problems in Mathematical Statistics.New York: Wiley
|
19 |
Kaas R, Goovaerts M, Dhaene J, Denuit M. Modern Actuarial Risk Theory Using R.2nd ed. Berlin; Springer, 2008
|
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
|
21 |
Liu Q, Lee J, Jordan M. A kernelized Stein discrepancy for goodness-of-fit tests. International Conference on Machine Learning, 2016, 276–284
|
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
|
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
|
24 |
Rao C R, Rubin H. On a characterization of the Poisson distribution. Sankhyā, 1964, 32(2-3): 295–298
|
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
|
26 |
Ramalingam S, Jagbir S. A characterization of the logarithmic series distribution and its application. Comm Statist Theory Methods, 1984, 13(7): 865–875
|
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
|
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
|
29 |
Shaked M, Shanthikumar J G. Stochastic Orders.Berlin: Springer, 2007
|
30 |
Shanbhag D N. An extension of the Rao-Rubin characterization of the Poisson distribution. J Appl Probab, 1977, 14(3): 640–646
|
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
|
32 |
Steutel F W. Preservation of Infinite Divisibility under Mixing and Related Topics. Math Centre Tracts, 33.Amsterdam: Mathematisch Centrum, 1970
|
33 |
Steutel F W, van Harn K. Infinite Divisibility of Probability Distributions on the Real Line.Boca Raton: CRC Press, 2003
|
34 |
Tsallis C. Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys, 1988, 52(1-2): 479–487
|
35 |
Temme N M. Special Functions: An Introduction to the Classical Functions of Mathematical Physics.New York: John Wiley & Sons, 2011
|
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
|
37 |
Willmot G E. The Poisson-inverse Gaussian distribution as an alternative to the negative binomial. Scand Actuar J, 1987, (3-4): 113–127
|
38 |
Wimmer G, Köhler R, Grotjahn R, Altmann G. Towards a theory of word length distribution. J Quant Linguist, 1994, 1(1): 98–106
|
39 |
Wimmer G, Altmann G. Thesaurus of Univariate Discrete Probability Distributions.Essen: Stamm, 1999
|
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
|
41 |
Zhang H, Li B. Characterizations of discrete compound Poisson distributions. Comm Statist Theory Methods, 2016, 45(22): 6789–6802
|
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
|
43 |
Zhang H, Jia J. Elastic-net regularized high-dimensional negative binomial regression: consistency and weak signals detection. arXiv: 1712.03412
|
44 |
Zygmund A. Trigonometric Series.Cambridge: Cambridge University Press, 2002
|
/
〈 | 〉 |