Permutation tests for stochastic ordering with ordered categorical data

Ordered categorical data are frequently encountered in many fields of research, such as, sociology, psychology, quality control, medical studies, and so forth. Especially in medical research, it is inevitable to meet a lot of problems containing ordered categorical data. Our specific interest is to find convincing solutions to some of the testing problems which include restrictions in the set of alternatives, such as testing for stochastic dominance and testing for monotonic stochastic ordering while using such a kind of data. When the number of nuisance parameters of underlying distributions or that of observed variables is small, there are some likelihood-based solutions. Our interest, however, is for cases where such numbers are not small. In these cases likelihood-based methods do not work, thus our interest is to proceed nonparametrically within permutation methods. Permutation methods are conditional on the pooled set of observed data which, in turn, are typically a set of sufficient statistics under the null hypothesis for the underlying distribution. Moreover, due to the evident complexity of such problems, according to Roy (1953), we also must use their Union-Intersection representation consisting on an equivalent break-down of the hypothesis under testing into a set of simpler sub-hypotheses for each of which a permutation test is available and such tests are jointly considered. So we must stay within the nonparametric combination of several dependent permutation tests. In the thesis, guided by two medical examples from the literature, we propose suitable solutions that are proved to be admissible combinations of optimal conditional partial tests and so enjoying good asymptotic properties.