A new weighted Poisson distribution for over- and under-dispersion situations. (English) Zbl 1524.62065
Summary: In this paper, we propose a four-parameter weighted Poisson distribution that includes and generalizes the weighted Poisson distribution proposed by Castillo and Pérez-Casany and the Conway-Maxwell-Poisson distribution, as well as other well-known distributions. It is a distribution that is a member of the exponential family and is an exponential combination formulation between the weighted Poisson distribution proposed by Castillo and Pérez-Casany and the Conway-Maxwell-Poisson distribution. This new distribution with an additional parameter of dispersion is more flexible, and the Fisher dispersion index can be greater than, equal to, or less than one. This last property allows it to model over-dispersed data as well as under-dispersed or equi-dispersed data. Many other properties of the new distribution are studied in this article. The parameters are estimated by two methods: the least squares and the maximum likelihood methods. The properties of the estimators are not studied because they follow directly. Two examples of application to real data, taking into account situations of overdispersion and underdispersion, are examined.
MSC:
| 62E10 | Characterization and structure theory of statistical distributions |
| 62E15 | Exact distribution theory in statistics |
| 62H10 | Multivariate distribution of statistics |
Keywords:
count data; over- and under-dispersion; exponential family; Conway-Maxwell-Poisson distribution; weighted Poisson distribution; estimationReferences:
| [1] | Castillo, J.D. and Pérez-Casany, M. (1998). Weighted Poisson distributions for overdispersion and underdispersion situations, Ann. Inst. Statist. Math., Vol. 50, No. 3, pp. 567-585. · Zbl 0912.62019 |
| [2] | Chakraborty, S. and Imoto, T. (2016). Extended Conway-Maxwell-Poisson distribution and its properties and applications, Journal of Statistical Distributions and Applications, Vol. 3, No. 5, pp. 1-19. doi:10.1186/s40488-016-0044-1 · Zbl 1351.62050 · doi:10.1186/s40488-016-0044-1 |
| [3] | Conway, R. and Maxwell, W. (1962). A queuing model with state dependent service rates, Journal of Industrial Engineering, Vol. 12, pp.132-136. |
| [4] | Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A. and Rubin, D.B. (1995). Bayesian Data Analysis, Third Edition, Chapman and Hall Book. |
| [5] | Greenwood, M. and Yule, U. (1920). An inquiry into the nature of frequency distributions represen-tative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents, J. Roy Statist. Soc. Ser. A, Vol. 83, pp. 255-279. |
| [6] | Gupta, P.L., Gupta, R.C. and Tripathi, R.C. (1997). On the monotonic properties of discrete failure rates, Journal of Statistical Planning and Inference, Vol. 65, pp. 255-268. · Zbl 0908.62099 |
| [7] | Gupta, R.C., Sim, S.Z. and Ong, S.H. (2014). Analysis of discrete data by Conway-Maxwell Pois-son distribution, AStA Adv Stat Anal, Vol. 98, pp. 327-343. · Zbl 1443.62045 |
| [8] | Henningsen, A. and Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R, Comput Stat, No. 26, pp. 443-458. · Zbl 1304.65039 |
| [9] | Huang, K. and Mi, J. (2018). Applications of likelihood ratio order in Bayesian inferences, Proba-bility in the Engineering and Informational Sciences, Vol. 34, No. 1, pp.1-13. · Zbl 07619764 |
| [10] | Kendall, M.G. (1961). Natural law in social sciences, Journal of the Royal Statistical Society Series A., Vol. 124., pp. 1-19. https://doi.org/10.2307/2984139 · doi:10.2307/2984139 |
| [11] | Kim, B., Kim, J., and Lee, S. (2018). Strong unimodality of discrete order statistics, Statistics & Probability Letters, Vol. 140, pp. 48-52. · Zbl 1463.62145 |
| [12] | Kokonendji, C.C., Mizère, D. and Balakrishnan, N. (2008). Connections of the Poisson weight function to overdispersion and underdispersion, Journal of Statistical Planning and Inference, No. 138, pp. 1287-1296. · Zbl 1133.62007 |
| [13] | Kulasekera, K.B. and Tonkyn, D.W. (1992). A new discrete distribution, with applications to sur-vival, dispersal and dispersion, Communications in Statistics -Simulation and Computation, Vol. 21, No. 2, pp 499-518. · Zbl 0850.62164 |
| [14] | Lakshmi, K., Mahaboob, B., Rajaiah, M. and Narayana, C. (2021). Ordinary least squares estima-tion of parameters of linear model, J. Math. Comput. Sci., Vol. 11, No. 2, pp. 2015-2030. |
| [15] | Louzayadio, C.G., Malouata, R.O. and Koukouatikissa Diafouka, M. (2021). A weighted Poisson distribution for underdispersed count data, International Journal of Statistics and Probability, Vol. 10, No. 4, pp. 157-165. |
| [16] | Meena, K.R. and Gangopadhyay, A.K. (2020). Estimating parameter of the selected uniform pop-ulation under the generalized stein loss function, Applications and Applied Mathematics: An International Journal (AAM), Vol. 15, Iss. 2, 894-915. · Zbl 1455.62046 |
| [17] | Minka, T.P., Shmueli, G., Kadane, J.B., Borle, S. and Boatwright, P. (2003). Computing with the COM-Poisson distribution, Technical Report 77. Department of Statistics, Carnegie Mellon University, Pittsburgh. (Available from http://www.stat.cmu.edu/tr/ |
| [18] | Patil, G.P. (2002). Weighted distributions, Encyclopedia of Environmetrics (eds. A.H. ElShaarawi and W.W. Piegorrsch), Vol. 4, pp. 2369-2377. Chichester: John Wiley & Sons. https://doi.org/10.1002/9781118445112.stat07359 · doi:10.1002/9781118445112.stat07359 |
| [19] | Patil, G.P., Rao, C.R. and Ratnaparkhi, M.V. (1986). On discrete weighted distributions and their use in model choice for observed data, Communications in Statistics-Theory and Methods, Vol. 15, No. 3, pp. 907-918. http://dx.doi.org/10.1080/03610928608829159 · Zbl 0601.62022 · doi:10.1080/03610928608829159 |
| [20] | Perrakis, S. (2019). Stochastic dominance option pricing: An alternative approach to option market research, Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-11590-6_1 · doi:10.1007/978-3-030-11590-6_1 |
| [21] | Qureshi, M.I. and Shadab, M. (2020). Some summation theorems for Clausen’s hypergeometric functions with unit argument, Applications and Applied Mathematics: An International Jour-nal (AAM), Vol. 15, Iss. 1, pp. 469-478. · Zbl 1444.33002 |
| [22] | Raeisi, M. and Yari, G. (2019). A further result on the aging properties of an extended additive hazard model, Applications and Applied Mathematics: An International Journal (AAM), Vol. 14, Iss. 2, pp. 777-783. · Zbl 1434.62214 |
| [23] | Rao, C.R. (1965). On discrete distributions arising out of methods of ascertainment, Sankhya Sr. A, pp. 311-324. · Zbl 0212.21903 |
| [24] | Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders, New York, NY: Springer New York. · Zbl 1111.62016 |
| [25] | Shmueli, G., Minka, T.P., Kadane, J.B., Borles, S. and Batwright, P. (2005). A useful distribution for fitting discrete data: Revival of the Conway-Maxwell-Poisson distribution, Appl. Statist., No. 54, pp. 127-142. · Zbl 1490.62058 |
| [26] | Weller, S.R. and Martin, J.H. (2020). On strongly unimodal third-order SISO linear systems with applications to pharmacokinetics, IFAC-PapersOnLine, Vol. 53, No. 2, pp. 4654-4661. |
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