  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.
