This proposed research applies a variety of mathematical techniques, including multivariate complex analysis and combinatorics, to open questions concerning inference from small samples. This inference includes standard frequentist techniques including the calculation of p-values and confidence intervals. The following aims are undertaken: 1. The approximation of DiCiccio and Martin (1993), conditional p-values using an approximate equivalence relation between a Bayesian credible region and a frequentist critical region, are improved, by determining an optimal or near-optimal set of initial conditions for the partial differential equation that is involved in the approximation. 2. Exact-enumeration techniques are applied to conditional inference in Cox regression. 3. An asymptotic approximation are constructed for the conditional distribution of a likelihood ratio statistic, under the regularity conditions far weaker then generally present in the existent literature.
Experimentalists routinely ask how well their data fits a hypothesis about the mechanism generating their data; the truth or falsehood of this hypothesis routinely is of importance to society as a whole. For example, in a medical clinical trial, one might investigate how closely data conform to a hypothesis that a new drug is equivalent at treating a particular disease to a standard drug, and in an engineering study, one might investigate how closely the data conform to a hypothesis that a part designed in a new way lasts no longer than a part designed in a conventional way. Disproving such a hypothesis often leads to adopting an alternative hypothesis that a new drug or part design actually represents an improvement. Disproving such a hypothesis involves a probabilistic proof by contradiction, in which investigators calculate the probability of observing data representing evidence at least as strong against the initial hypothesis, and reject this hypothesis if this probability is small. For example, investigators interested in a new design for a low-emission vehicle might compare a new battery design to an old design, in an experiment in which batteries of both types are installed in vehicles and tested under a variety of conditions. In situations like this, the method for quantifying evidence against the hypothesis of equal reliability of both batteries is well established. This research helps to attach probabilities to the various possible outcomes of the experiment under various assumptions about the relative reliabilities of the batteries, accounting for variation in experimental conditions and for various non-battery reasons for vehicle failure. Similar questions arise in medicine, finance, the social sciences, and many other fields of interest.
