Abstract
Signed-root formulae were first discussed by Lawley (1956), as a method for approximate confidence intervals in frequentist inference. Since then, they have been pursued by many authors, in both frequentist and Bayesian inference. Sweeting (1995, 1996) provides a unified framework for consideration of signed-root asymptotics, incorporating results such as the approximate posterior marginal probabilities of DiCiccio et al. (1990) and the transformed signed-root, proposed by Barndorff-Nielsen (1983, 1988) in a frequentist context. This thesis follows on from the work of Sweeting (1995, 1996), and the Bayesian results in particular. The emphasis is on new applications, rather than new asymptotic results. A mixture modelling approach is used to allow application of signed-root results in sub-asymptotic cases - for example, multimodal likelihoods. Although the real value of asymptotic methods is in multi-parameter problems, the single-parameter approach is considered first for each method in order to illustrate the approach. Although rigorous asymptotic proofs are, by definition, not avail-able in these sub-asymptotic applications, some justification is given and trials suggest that the approach is sound. The power and the limitations of the methods are explored with reference to the bivariate normal example of Efron (1975) and the hematopoiesis data of Newton et al. (1995). In addition, the thesis aims to be a "user's manual" for Bayesian asymptotics: as well as a comprehensive review of the literature, issues of implementation and computing are addressed in detail in the penultimate chapter.