This paper discusses the use of regression models of count data. It presents the estimation of the parameters for Poisson regression and zero-truncated Poisson regression models using the maximum likelihood method. We are interested in studying the performance of the estimators of modified Poisson regression models of count data. An assessment of the maximum likelihood estimates of the parameters is presented through a numerical study for different sample sizes for each model. Two empirical applications for non-truncated and truncated count data are presented, the first studies the effect of wave damage to cargo-ship and the second application investigates the length of hospital stay in days.