The inverse trinomial distribution and its application in modeling over-dispersed count Data

Document Type : Original Scientific Paper

Author

LOYAB Statistical Consulting & Data Analytics, Adeleke Road, Lamodi, Offa. Kwara State, Nigeria

Abstract

In this paper, we apply the inverse trinomial distribution with parameters (p, q, λ) to count data exhibiting over-dispersion. We compare results for the mean parameterized inverse trinomial and the parameter λ as linear predictors in the generalized linear model case. Our results also demonstrate methods for obtaining the means and variances for the zero-inflated and zero-truncated inverse trinomial distributions. Results obtained here indicate that the generalized Poisson type II has a close relationship with the inverse trinomial distributions. Several data examples are employed in this paper for both frequency and data having covariates cases. SAS PROC NLMIXED is employed using adaptive Gaussian quadrature and the Newton-Raphson as the optimizer.

Keywords

Main Subjects

References

Adesina, O.S., Agunbiade, D.A. and Oguntunde, P.E. (2021). Flexible Bayesian Dirichlet mixtures of generalized linear mixed models for count data. Scientific African, 13:1–11.
Chakraborty, S. and Imoto, T. (2016). Extended Conway-Maxwell-Poisson distribution and its properties and applications. Journal of Statistical Distributions and Applications, 3(5):1–199.
Chey, V.K. (2002). Comparison of moth diversity between lightly and heavily logged sites in a tropical rain forest. Malayan Nature Journal, 56(1):23–41.
Consul, P.C. (1989). Generalized Poisson Distributions; Properties and Applications. New York: Marcel Dekker.
Consul, P.C. and Famoye, F. (1992). Generalized Poisson Regression Model. Communications in Statistics-Theory and Methods, 21(1):89–109.
Deb, P. and Trivedi, P.K. (1997). Demand for medical care by the elderly: A finite mixture approach. Journal of Applied Econometrics, 12:313–336.
Flynn, M. and Francis, L.A. (2009). More flexible GLMs zero-inflated models and hybrid models. Casualty Actuarial Society, 2009:148–224.
Hilbe, J.M. (2013). Negative Binomial Regression. Cambridge University Press.
Joe, H. and Zhu, R. (2005). Generalized Poisson distribution: The property of mixture of Poisson and comparison with negative binomial distribution. Biometrical Journal: Journal of Mathematical Methods in Biosciences, 47(2):219–229.
Khang, T.F. and Ong, S.H. (2007). A new generalization of the logarithmic distribution arising from the inverse trinomial distribution. Communications in Statistics-Theory and Methods, 36:3–21.
Lawal, H.B. (1980). Tables of percentage points of Pearson's goodness-of-fit statistic for use with small expectations. Applied Statistics, 29:292–298.
Lawal, H.B. (2017). On the negative binomial-generalized exponential distribution and its applications. Applied Mathematical Sciences, 11:345–360.
Lawal, H.B. (2018). Correcting for non-sum to 1 estimated probabilities in applications of discrete probability models to count data. International Journal of Statistics and Probability, 6:119–131.
Lawal, H.B. (2019). Quasi-negative binomial, inverse trinomial and negative binomial generalized exponential distributions and their applications. Journal of Probability and Statistical Science, 17(2):205–221.
Letac, G. and Mora, M. (1990). Natural real exponential families with cubic variance functions. Annals of Statistics, 18:1–37.
Li, S., Yang, F., Famoye, F., Lee, C. and Black, D. (2011). Quasi-negative binomial distribution: Properties and applications. Computational Statistics and Data Analysis, 55(20):2363–2371.
Phang, Y.N. and Ong, S.H. (2006). Zero-Inflated inverse trinomial distribution for modeling count data. Proceedings, 2nd IMT-GT Regional Conference.
Phang, Y.N. and Loh, E.F. (2014). Statistical analysis for overdispersed medical count data. International Scholarly and Scientific Research, 8(2):292–294.
Phang, Y.N., Sim, S.Z. and Ong, S.H. (2013). Statistical analysis for the inverse trinomial distribution. Communications in Statistics-Simulation and Computation, 42(9):2073–2085.
Pinheiro, J.C. and Bates, D.M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4:12–35.
Shimizu, K. and Yanagimoto, T. (1991). The inverse trinomial distribution. Japan Journal of Applied Statistics, 20:89–96.
Yanagimoto, T. (1989). The inverse binomial distribution as a statistical model. Communications in Statistics: Theory and Methods, 18(10):3629–3633.
Journal of Statistical Modelling: Theory and Applications
Volume 2, Issue 2
July 2021
Pages 25-39

Files

History

  • Receive Date: 13 December 2021
  • Revise Date: 21 February 2022
  • Accept Date: 23 February 2022

Share

How to cite

Statistics