Higher order asymptotic refinements in Bell regressions

The discrete Bell distribution and its associated regression model were introduced recently in the statistical literature. The Bell distribution has proved to be a useful alternative to the traditional Poisson distribution, mainly to deal with overdispersion. Likelihood‐based inference on the Bell regression parameters relies on asymptotic assumptions like the sample size going to infinity. In this paper, we focus on the small‐sample case and consider higher order asymptotic refinements in this class of regression models. In particular, we derive the second‐order biases of the maximum likelihood estimators, which are used to define bias‐corrected estimators. The preventive method to bias reducing is also considered. We provide a simple matrix formula for the skewness of the distributions of the maximum likelihood estimators, and also derive a simple matrix formula for the second‐order variance–covariance matrix of the maximum likelihood estimators in this class of regression models. We use Monte Carlo simulations to verify the performance of the proposed methods. Our simulation results suggest that the analytical expressions we derive deliver more reliable results in small‐sized samples. An empirical application to a real data set is considered for illustrative purposes.