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被广泛使用。然而，当真实数据生成过程为泊松分布时，往往难以将其与具有极端参数取值的负二项分布区分开来，且现有针对负二项分布的最大似然估计程序可能失效或产生不稳定的估计值。为解决这一问题，我们开发了一种新算法用于计算负二项分布参数的最大似然估计，该算法比现有方法更高效、更精确。我们进一步通过引入新的参数化方式扩展了负二项分布，使其将泊松分布作为特例涵盖其中。我们提供的理论论证表明：当应用于泊松分布数据时，扩展负二项分布的参数估计能够一致地恢复真实的泊松分布。