On the Bell distribution and its associated regression model for count data

•We define the one-parameter discrete Bell distribution, which has a simple form for its probability mass function.•It is shown to be a particular solution of a multiple Poisson process.•It is useful for modeling count data with greater variability than the Poisson distribution allows.•We introduce a Bell regression model with statistical inference procedures, diagnostic tools, and computer implementation.•An application to real data suggests that the new model is useful in practice. In this paper we define and study the one-parameter discrete Bell distribution, which has a very simple form for its probability mass function. In particular, we show that this discrete distribution is a particular solution of a multiple Poisson process. By considering the distribution studied in this article, we introduce a new regression model where the response variable is a count. Further, residuals are also proposed for the new count regression model. An empirical application is considered to show the usefulness of the Bell regression model in practice.