  This work proposes a new method for computing acceptance regions of exact
multinomial tests. From this an algorithm is derived, which finds exact
p-values for tests of simple multinomial hypotheses. Using concepts from
discrete convex analysis, the method is proven to be exact for various popular
test statistics, including Pearson's chi-square and the log-likelihood ratio.
The proposed algorithm improves greatly on the naive approach using full
enumeration of the sample space. However, its use is limited to multinomial
distributions with a small number of categories, as the runtime grows
exponentially in the number of possible outcomes.
  The method is applied in a simulation study and uses of multinomial tests in
forecast evaluation are outlined. Additionally, properties of a test statistic
using probability ordering, referred to as the "exact multinomial test" by some
authors, are investigated and discussed. The algorithm is implemented in the
accompanying R package ExactMultinom.
