Negative binomial related distributions have been widely used in practice. The calculation of the corresponding Fisher information matrices involves the expectation of trigamma function values which can only be calculated numerically and approximately. In this paper, we propose a trigamma-free approach to approximate the expectations involving the trigamma function, along with theoretical upper bounds for approximation errors. We show by numerical studies that our approach is highly efficient and much more accurate than previous methods. We also apply our approach to compute the Fisher information matrices of zero-inflated negative binomial (ZINB) and beta negative binomial (ZIBNB) probabilistic models, as well as ZIBNB regression models.

翻译：负二项相关分布在实际应用中广泛使用。计算相应的Fisher信息矩阵涉及对三伽玛函数值的期望，而这些值只能通过数值方法近似计算。本文提出了一种免三伽玛函数的方法来近似计算包含三伽玛函数的期望，并给出了近似误差的理论上界。通过数值研究，我们证明该方法相比现有方法具有更高的计算效率和更高的精度。我们还将该方法应用于计算零膨胀负二项（ZINB）和贝塔负二项（ZIBNB）概率模型的Fisher信息矩阵，以及ZIBNB回归模型的Fisher信息矩阵。