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：负二项相关分布已在实践中得到广泛应用。计算对应的Fisher信息矩阵涉及三角伽马函数值的期望，这些期望仅能通过数值方法近似计算。本文提出了一种免三角伽马函数的方法来近似涉及三角伽马函数的期望，并给出了近似误差的理论上界。数值研究表明，该方法效率高且精度远超现有方法。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）概率模型以及ZIBNB回归模型的Fisher信息矩阵。