A distribution arising from a random walk on the plane / Sim Shin Zhu - Sim, Shin Zhu (2011)
This thesis proposed a new distribution to model under-, equi- and overdispersion in count data. It arises as a particular case of a modified random walk on the plane, and includes the binomial, negative binomial and shifted negative binomial as special cases. Some properties, test of hypothesis for equi-dispersion, simulation study of power, parameter estimation by maximum likelihood and a squared distance method based on the probability generating function are considered. The proposed distribution is compared with existing distributions like the COM-Poisson and generalized Poisson for modelling dispersion. It is shown to be a flexible model in applications by illustrating its goodness-of-fit to four real data sets. In addition, two new bivariate distributions are derived from proposed distribution as the alternative bivariate discrete distributions by applying the convolution of two bivariate distributions and the classical trivariate reduction method. It is found that the new bivariate distributions permit more flexibility in modelling and less limitation on the correlation between the two random variables. The characteristic and the estimation of the parameters of the new bivariate distributions are provided. Furthermore, the statistical inference for the COM-Poisson distribution and computational issues are studied. Test for equi-dispersion, study of statistical power and parameter estimation by maximum likelihood are developed. This thesis also considers a probability generating function-based divergence statistic for parameter estimation. The performance and robustness of the proposed statistic in parameter estimation is studied for the negative binomial distribution by Monte Carlo simulation, especially in comparison with maximum likelihood and minimum Hellinger distance estimation.