From Poisson Observations to Fitted Negative Binomial Distribution

The Kolmogorov-Smirnov (KS) test has been widely used for testing whether a random sample comes from a specific distribution, possibly with estimated parameters. If the data come from a Poisson distribution, however, one can hardly tell that they do not come from a negative binomial distribution by running a KS test, even with a large sample size. In this paper, we rigorously justify that the KS test statistic converges to zero almost surely, as the sample size goes to infinity. To prove this result, we demonstrate a notable finding that in this case the maximum likelihood estimates (MLE) for the parameters of the negative binomial distribution converge to infinity and one, respectively and almost surely. Our result highlights a potential limitation of the KS test, as well as other tests based on empirical distribution functions (EDF), in efficiently identifying the true underlying distribution. Our findings and justifications also underscore the importance of careful interpretation and further investigation when identifying the most appropriate distributions in practice.