  In this paper, we address the problem of testing goodness-of-fit for discrete
distributions, where we focus on the geometric distribution. We define new
likelihood-based goodness-of-fit tests using the beta-geometric distribution
and the type I discrete Weibull distribution as alternative distributions. The
tests are compared in a simulation study, where also the classical
goodness-of-fit tests are considered for comparison. Throughout the paper we
consider conditional testing given a minimal sufficient statistic under the
null hypothesis, which enables the calculation of exact p-values. For this
purpose, a new method is developed for drawing conditional samples from the
geometric distribution and the negative binomial distribution. We also explain
briefly how the conditional approach can be modified for the binomial, negative
binomial and Poisson distributions. It is finally noted that the simulation
method may be extended to other discrete distributions having the same
sufficient statistic, by using the Metropolis-Hastings algorithm.
