Normal Approximations for Estimating the Differences and Ratios of Parameters and Means of two Zero-Truncated Poisson Distributions

Count data, a common type of data in applied research, does not always include zero values. In such cases, the Poisson distribution is not suitable, and a Zero-Truncated Poisson distribution should be considered instead. Deriving closed-form expressions for the Method of Moments estimators for the parameters and means of Zero-Truncated Poisson distributions appears infeasible. In this work, we focus on the case of two Zero-Truncated Poisson distributions and propose novel estimators for the differences and ratios of their parameters and means. Their asymptotic normality is then examined using the traditional Delta method, and simulations are used to assess their performance. An earthquake dataset (worldwide number of events of magnitude \(\geq\) 7.5 over 1984–2023) serves as a real-life example to demonstrate the application of these new estimators.