Exact Conditional Tests and Approximate Bootstrap Tests for the von Mises Distribution

Exact and approximate tests of fit are compared for testing that a given sample comes from the von Mises distribution. For the exact test, Gibbs sampling is used to generate samples from the conditional distribution of sample data, given the values of the sufficient statistics. The samples, called co-sufficient samples, are used to estimate the distribution of Watson's statistic, and hence to find the exact p-value for the given sample. The test is compared to the approximate test using the parametric bootstrap. Two examples are analyzed, and the p-values of the two tests are compared. When more examples are examined, an unexpectedly high correlation is discovered between the two sets of p-values, suggesting a strong mathematical connection.