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.

中文翻译：

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Bell 回归中的高阶渐近细化

最近在统计文献中引入了离散贝尔分布及其相关的回归模型。贝尔分布已被证明是传统泊松分布的有用替代方案，主要用于处理过度离散。对贝尔回归参数的基于似然的推断依赖于样本量趋于无穷大等渐近假设。在本文中，我们关注小样本情况，并考虑此类回归模型中的高阶渐近细化。特别是，我们推导出最大似然估计量的二阶偏差，用于定义偏差校正估计量。还考虑了减少偏差的预防方法。我们为最大似然估计量分布的偏度提供了一个简单的矩阵公式，并且还推导了此类回归模型中最大似然估计量的二阶方差-协方差矩阵的简单矩阵公式。我们使用蒙特卡罗模拟来验证所提出方法的性能。我们的模拟结果表明，我们推导出的分析表达式在小样本中提供了更可靠的结果。出于说明目的考虑对真实数据集的经验应用。