Most powerful rank tests for perfect rankings

We consider the problem of testing for perfect rankings in ranked set sampling (RSS). By using a new algorithm for computing the probability that specified independent random variables have a particular ordering, we find most powerful rank tests of the null hypothesis of perfect rankings against fully specified alternatives. We compare the power of these most powerful rank tests to that of existing rank tests in the literature, and we find that the existing tests are surprisingly close to optimal over a wide range of alternatives to perfect rankings. This finding holds both for balanced RSS and for unbalanced RSS cases where the different ranks are not equally represented in the sample. We find that the best of the existing tests is the test that rejects when the null probability of the observed ranks is small, and we provide a new, more efficient R function for computing the test statistic.