On goodness of fit tests for the Poisson, negative binomial and binomial distributions

In this paper, we address the problem of testing the fit of three discrete distributions, giving a brief account of existing tests and proposing two new tests. One of the new tests is for any discrete distribution function. This general test is a discrete version of a recently proposed test for the skew-normal in Potas et al. (Appl Math Sci 8(78):3869-3887, 2014), which in turn is based on a test for normality in Zhang (J R Stat Soc Ser B 64(2):281-294, 2002). The other test which is proposed is given explicitly for testing the Poisson, the negative binomial or the binomial distributions. It is based on earlier work by Gonzalez-Barrios et al. (Metrika 64:77-94, 2006) done for distributions within the power series family. This test uses the conditional density (probability) of the observed sample, given the value of the minimal sufficient statistic. The proposed new test is defined as a conditional probabilities ratio, modifying the previous criterion. An extensive simulation study to compare the power of the proposed new tests with established tests of fit is carried out, using alternatives that had been used in previous simulations, for these particular distributions. In the study, all tests compared were used identifying their exact conditional distribution given the adequate sufficient statistic, so no approximations were made by using asymptotic distributions and possibly estimates of the unknown parameters present in their asymptotic distributions.