From Poisson Observations to Fitted Negative Binomial Distribution

Negative binomial distribution has been widely used as a more flexible model than Poisson distribution for count data. When the observations come from a Poisson distribution, it is often challenging to rule out the possibility that the data come from a negative binomial distribution with extreme parameter values. To address this phenomenon, we develop a more efficient and accurate algorithm for finding the maximum likelihood estimate of negative binomial parameters, which outperforms the state-of-the-art programs for the same purpose. We also theoretically justify that the negative binomial distribution with parameters estimated from Poisson data converges to the true Poisson distribution with probability one. As a solution to this phenomenon, we extend the negative binomial distributions with a new parameterization, which include Poisson distributions as a special class.