On Zero-Modified Poisson-Sujatha Distribution to Model Overdispersed Count Data
Abstract
In this paper we propose the zero-modified Poisson-Sujatha distribution as an alternative to model overdispersed count data exhibiting inflation or deflation of zeros. It will be shown that the zero modification can be incorporated by using the zero-truncated Poisson-Sujatha distribution. A simple reparametrization of the probability function will allow us to represent the zero-modified Poisson-Sujatha distribution as a hurdle model. This trick leads to the fact that proposed model can be fitted without any previously information about the zero modification present in a given dataset. The maximum likelihood theory will be used for parameter estimation and asymptotic inference concerns. A simulation study will be conducted in order to evaluate some frequentist properties of the developed methodology. The usefulness of the proposed model will be illustrated using real datasets of the biological sciences field and comparing it with other models available in the literature.
References
Angers JF, Biswas A (2003). "A Bayesian analysis of zero–inflated generalized Poisson model". Computational statistics & data analysis, 42(1), 37–46.
Bahn GD, Massenburg R (2008). "Deal with Excess Zeros in the Discrete Dependent Variable, the Number of Homicide in Chicago Census Tract". In Joint Statistical Meetings of the American Statistical Association, pp. 3905–12.
Beall G (1940). "The fit and significance of contagious distributions when applied to observations on larval insects". Ecology, 21(4), 460–474.
Beuf KD, Schrijver JD, Thas O, Criekinge WV, Irizarry RA, Clement L (2012). "Improved base–calling and quality scores for 454 sequencing based on a Hurdle Poisson model". BMC bioinformatics, 13(1), 303.
Bohara AK, Krieg RG (1996). "A zero–inflated Poisson model of migration frequency". International Regional Science Review, 19(3), 211–222.
Cancho VG, Louzada Neto F, Barriga GDC (2011). "The Poisson–Exponential lifetime distribution". Computational Statistics & Data Analysis, 55(1), 677–686.
Conceição KS, Andrade MG, Louzada Neto F (2013). "Zero–modified Poisson model: Bayesian approach, influence diagnostics, and an application to a Brazilian leptospirosis notification data". Biometrical Journal, 55(5), 661–678.
Dietz E, Bohning D (2000). "On estimation of the Poisson parameter in zero–modified Poisson models". Computational Statistics & Data Analysis, 34(4), 441–459.
Ghitany ME, Al-Mutairi DK, Nadarajah S (2008). "Zero–truncated Poisson–Lindley distribution and its application". Mathematics and Computers in Simulation, 79(3), 279–287.
Gurmu S, Trivedi PK (1996). "Excess zeros in count models for recreational trips". Journal of Business & Economic Statistics, 14(4), 469–477.
Heilbron DC, Gibson DR (1990). "Shared needle use and health beliefs concerning AIDS: regression modeling of zero–heavy count data. Poster session". In Proceedings of the Sixth International Conference on AIDS, San Francisco, CA.
Hörmann W, Leydold J, Derflinger G (2013). Automatic nonuniform random variate generation. Springer Science & Business Media.
Hu MC, Pavlicova M, Nunes EV (2011). "Zero–inflated and hurdle models of count data with extra zeros: examples from an HIV–risk reduction intervention trial". The American journal of drug and alcohol abuse, 37(5), 367–375.
Lambert D (1992). "Zero–inflated Poisson regression, with an application to defects in manufacturing". Technometrics, 34(1), 1–14.
Loeschcke V, Köhler W (1976). "Deterministic and stochastic models of the negative binomial distribution and the analysis of chromosomal aberrations in human leukocytes". Biometrische Zeitschrift, 18(6), 427–451.
McDowell A (2003). "From the help desk: hurdle models". The Stata Journal, 3(2), 178–184.
Mullahy J (1986). "Specification and Testing of Some Modified Count Data Models". Journal of Econometrics, 91(434), 841–853.
R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL: http://www.R-project.org.
Ridout M, Dem ́etrio CGB, Hinde J (1998). "Models for count data with many zeros". In Proceedings of the XIXth international biometric conference, volume 19, pp. 179–192.
Saffar SE, Adnan R, Greene W (2012). "Parameter estimation on hurdle Poisson regression model with censored data". Jurnal Teknologi, 57(1), 189–198.
Sankaran M (1970). "The Discrete Poisson–Lindley Distribution". Biometrics, 26(1), 145–149.
Shanker R (2015). "Sujatha distribution and Its Applications". Statistics in Transition new Series.
Shanker R (2016). "The Discrete Poisson–Sujatha Distribution". International Journal of Probability and Statistics, 5(1), 1–9.
Shanker R, Fesshaye H (2016a). "On Poisson–Sujatha Distribution and its Applications to Model Count Data from Biological Sciences". Biometrics & Biostatistics International Journal, 3(5), 1–13.
Shanker R, Fesshaye H (2016b). "Size–Biased Poisson Sujatha distribution with Applications". Communicated.
Shanker R, Fesshaye H (2016c). "Zero–Truncated Poisson–Sujatha distribution with Applications". Communicated.
Shanker R, Fesshaye H (2016d). "On Zero–Truncation of Poisson, Poisson–Lindley and Poisson–Sujatha Distributions and their Applications". Biometrics & Biostatistics International Journal, 3(5).
Shanker R, Mishra A (2013). "A quasi Lindley distribution". African Journal of Mathematics and Computer Science Research, 6(4), 64–71.
Zamani H, Ismail N (2010). "Negative Binomial–Lindley Distribution and Its Application". Journal of Mathematics and Statistics, 6(1), 4–9.
Zorn CJW (1996). "Evaluating zero–inflated and hurdle Poisson specifications". Midwest Political Science Association, 18(20), 1–16.
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