This thesis discusses various issues in the estimation of models for count data. In the first part of the thesis, we derive an analytic expression for the bias of the maximum likelihood estimator (MLE) of the parameter in a doubly-truncated Poisson distribution, which proves highly effective as a means of bias correction. We explore the circumstances under which bias is likely to be problematic, and provide some indication of the statistical significance of the bias. Over a range of sample sizes, our method outperforms the alternative of bias correction via the parametric bootstrap. We show that MLEs obtained from sample sizes which elicit appreciable bias also have sampling distributions which are unsuited to be approximated by large-sample asymptotics, and bootstrapping confidence intervals around our bias-adjusted estimator is preferred, as two tiers of bootstrapping may incur a heavy computational burden. Modelling count data where the counts are strictly positive is often accomplished using a positive Poisson distribution. Inspection of the data sometimes reveals an excess of ones, analogous to zero-inflation in a regular Poisson model. The latter situation has well developed methods for modelling and testing, such as the zero-inflated Poisson (ZIP) model, and a score test for zero-inflation in a ZIP model. The issue of count inflation in a positive Poisson distribution does not seem to have been considered in a similar way. In the second part of the thesis, we propose a one-inflated positive Poisson (OIPP) model, and develop a score test to determine whether there are “too many” ones for a positive Poisson model to fit well. We explore the performance of our score test, and compare it to a likelihood ratio test, via Monte Carlo simulation. We find that the score test performs well, and that the OIPP model may be useful in many cases. The third part of the thesis considers the possibility of one-inflation in zero-truncated data, when overdispersion is present. We propose a new model to deal with such a phenomenon, the one-inflated zero-truncated negative binomial (OIZTNB) model. The finite sample properties of the maximum likelihood estimators for the parameters of such a model are discussed. This Chapter considers likelihood ratio tests which assist in specifying the OIZTNB model, and investigates the finite sample properties of such tests. The OIZTNB model is illustrated using the medpar data set, which describes the hospital length of stay for a set of patients in Arizona. This is a data set that is widely used to highlight the merits of the zero-truncated negative binomial (ZTNB) model. We find that our OIZTNB model fits the data better than does the ZTNB model, and this leads us to conclude that the data are generated by a one-inflated process. / Graduate