The negative binomial distribution has been widely used as a more flexible model than the Poisson distribution for count data. However, when the true data-generating process is Poisson, it is often challenging to distinguish it from a negative binomial distribution with extreme parameter values, and existing maximum likelihood estimation procedures for the negative binomial distribution may fail or produce unstable estimates. To address this issue, we develop a new algorithm for computing the maximum likelihood estimate of negative binomial parameters, which is more efficient and more accurate than existing methods. We further extend negative binomial distributions with a new parameterization to cover Poisson distributions as a special class. We provide theoretical justifications showing that, when applied to a Poisson data, the estimated parameters of the extended negative binomial distribution can consistently recover the true Poisson distribution.

翻译：负二项分布作为一种比泊松分布更灵活的计数数据模型已被广泛应用。然而，当真实数据生成过程为泊松分布时，通常难以将其与参数值极端的负二项分布区分开，且现有负二项分布的最大似然估计方法可能失效或产生不稳定估计。为解决此问题，我们开发了一种计算负二项分布参数最大似然估计的新算法，其效率与精度均优于现有方法。我们进一步通过新参数化扩展了负二项分布，使其将泊松分布作为特例涵盖。理论证明表明，当应用于泊松数据时，扩展负二项分布的估计参数能够一致地还原真实泊松分布。