The Touchard Process for Count Data with Dependent Increments

This paper introduces the Touchard process, a flexible two-parameter stochastic framework for modeling count data that depart from the classical Poisson assumptions. In contrast to standard Poisson processes, the proposed model allows for both nonstationary and dependent increments, enabling the representation of overdispersion, underdispersion, and temporal dependence within a unified structure. The main contribution lies in extending weighted Poisson models to a stochastic-process setting through recursively defined transition probabilities associated with Touchard marginal distributions. We derive key theoretical properties, including admissibility conditions and a recursive formulation for the transition probabilities, and propose an efficient simulation algorithm. Maximum likelihood estimation is developed for parameter inference, and a likelihood ratio framework is used for model comparison. An empirical application to motor vehicle crash data illustrates the ability of the model to capture dynamic patterns that are not adequately described by classical Poisson-based approaches.