An Exact Kolmogorov–Smirnov Test for the Negative Binomial Distribution with Unknown Probability of Success

Abstract

Abstract
In this paper, we develop an exact Kolmogorov–Smirnov goodness-of-fit test in the case of negative binomial distribution with an unknown probability of success. This test is conditional, with the test statistic being the maximum absolute difference between the empirical distribution function and its conditional expectation given the sample total. The results are not asymptotic, but exact. We illustrate the test with three examples in case the size parameter equals one i.e. the geometric distribution. We also include some simulations in order to check the power of the procedures. The new test seems to be the first exact negative binomial goodness-of-fit test for which critical values are available without simulation or exhaustive enumeration.

Keywords: conditional test, Cramér–von Mises statistics, Anderson-Darling statistics, goodness of fit