Unified Frequentist and Bayesian testing of precise hypotheses: In fixed samples and sequential settings

The major problem faced in the field of statistics is the deep schism between the Frequentists and the Bayesians, particularly in regard to testing statistical hypotheses and their relevant interpretations. In this thesis, we propose a viable resolution to the conflict. Following the proposal of Berger, Brown and Wolpert (1994), we develop new statistical procedures which allow a synthesis and an agreement between the Bayesian\u27s and the Frequentist\u27s approaches to testing precise hypothesis in fixed samples and in sequential settings. The new testing procedures have, simultaneously, a valid Bayesian interpretation and a valid (conditional) Frequentist interpretation. More specifically, we propose a conditioning strategy under which, both the Conditional Frequentist and the Bayesian who utilize this new testing procedure, will report the same error probabilities upon rejecting or accepting. This is of considerable interest because it is often perceived that Bayesian and Frequentist testing are incompatible in this situation. In this work, we have begun the lengthy, and rather complex task of \u27re-tooling\u27 testing procedures. We provide some \u27default\u27 tests for practitioners to use in common situations. We also demonstrate the application of the new testing procedures even in such sequential settings as those of sequential clinical trials. The resulting testing procedures do not depend on the stopping rule employed. This is a startling indication in that, even in the area of sequential testing, Frequentists and Bayesians can now arrive at an agreement on the final analysis