In the statistical analysis of count data an important characteristic to examine is the equality of the mean and the variance (equi-dispersion). Observed data tends to exhibit unequal mean and variance making the equi-dispersed Poisson distribution, a popular model for count data, untenable. The case of the variance exceeding the mean (over dispersion) is prevalent in many areas. Therefore various approaches have been used to extend and modify the Poisson distribution.
Notable approaches are those by birth process modelling (Faddy, 1997), mixtures leading to over dispersed distributions (variance greater than the mean) and weighting of Poisson distribution (Castillo and Perez-Casany, 1998). Winkelmann (1995) examined an approach using the theory of renewal processes to derive a generalized model. There are few distributions in the literature that are able to represent under, equi- and over dispersion.
In this talk a new distribution that arises as a particular case of a modified random walk on the plane is considered which can model under, equi and over dispersion in count data. Next further developments of the COM-Poisson distribution (Shmueli etal, 2005), another distribution which can also model under, equi and over dispersion, are discussed. Following Winkelmann’s approach, a distribution, with generalized Weibull duration in the renewal process, for modelling over dispersion is presented with a discussion on its applications.
References
Castillo JD, Perez-Casany M (1998). Weighted Poisson distributions for over dispersion and under-dispersion situations. Ann Inst Statist Math 50 (3): 567--585.
Faddy MJ (1997). Extending Poisson process modelling and analysis of count data. Biometrical Journal 39(4): 431--440.
Shmueli G, Borle S and Boatwright P (2005). A useful distribution for fitting discrete data: revival of the Conway-Maxwell-Poisson distribution.
J Roy Statist Soc C 54: 127--142.
Winkelmann R (1995). Duration dependence and dispersion in count-data models. Journal of Business and Economic Statistics 13(4): 467--447.