Statistical inference for negative binomial processes with applications to market research

The negative binomial distribution (NBD) and negative binomial processes have been used as natural models for events occurring in fields such as accident proneness accidents and sickness market research insurance and risk theory. The fitting of negative binomial processes in practice has mainly focussed on fitting the one-dimensional distribution, namely the NBD, to data. In practice, the parameters of the NBD are usually estimated by using inefficient moment based estimation methods due to the ease in estimating moment based estimators in comparison to maximum likelihood estimators. This thesis develops efficient moment based estimation methods for estimating parameters of the NBD that can be easily implemented in practice. These estimators, called power method estimators, are almost as efficient as maximum likelihood estimators when the sample is independent and identically distributed. For dependent NBD samples, the power method estimators are more efficient than the commonly used method of moments and zero term method estimators. Fitting the one-dimensional marginal distribution of negative binomial processes to data gives partial information as to the adequacy of the process being fitted. This thesis further develops methods of statistical inference for data generated by negative binomial processes by comparing the dynamical properties of the process to the dynamical properties of data. For negative binomial autoregressive processes, the dynamical properties may be checked by using the autocorrelation function. The dynamical properties of the gamma Poisson process are considered by deriving the asymptotic covariance and correlation structures of estimators and functionals of the gamma Poisson process and verifying these structures against data. The adequacy of two negative binomial processes, namely the gamma Poisson process and the negative binomial first-order autoregressive process, as models for consumer buying behavior are considered. The models are fitted to market research data kindly provided by ACNielsen BASES