Assistant Professor, Department of MathematicsISEG, Technical University of Lisbon, Lisbon, Portugal
Problems on the Bayesian/Frequentist Interface
Two areas on the Bayesian/frequentist interface are explored. The first is simultaneous Bayesian-conditional frequentist hypotheses testing; the second is the development of objective prior distributions for the parameters of Gaussian spatial processes. After presenting a structured overview of the theory of unified Bayesian-conditional frequentist testing, three particular problems in the area are explored in depth. The first is that of testing a simple hypothesis concerning the mean of an Exponentially distributed population, while the second is that of comparing the means of two Exponential random variables. In each setting, two different objective priors, leading to two different Bayesian-conditional frequentist tests, are considered and the presence of censoring and its consequences are investigated. The third problem considered in this area is that of sequentially testing a precise hypothesis concerning the drift of a Brownian motion observed continuously in time. Most of the commonly used stopping boundaries do not conform to the established theory of conditional frequentist testing. Hence, a new conditioning strategy is developed, considerably extending the existing approaches. We first study the sequential probability ratio test for simple hypotheses, and then generalize to quite arbitrary stopping boundaries, including vertical lines. In the second part of the dissertation and motivated by the statistical evaluation of complex computer models, we deal with the issue of objective prior specification for the parameters of Gaussian spatial processes. In particular, we derive the Jeffreys-rule, Jeffreys independence and reference priors for this situation, and prove that the resulting posterior distributions are proper under a quite general set of conditions. Another prior specification strategy, based on maximum likelihood estimates, is also considered, and all priors are then compared on the grounds of the frequentist properties of the ensuing Bayesian procedures.