Dr Glen Livingston

The Kruskal¿Wallis tests are appropriate tests for the completely randomised design, both for when the data are untied ranks, and, with adjustment, for when there are ties and mid-ranks are used. Both these tests are shown to be Cochran¿Mantel¿Haenszel mean score tests. The relationship between the Kruskal¿Wallis test statistic and the ANOVA F test statistic when there are no ties generalises to the same relationship between the Cochran¿Mantel¿Haenszel mean score test statistic and the ANOVA F test statistic. It thus also relates both Kruskal¿Wallis test statistics to the ANOVA F test statistic. A small simulation study finds that p-values may be more accurately found using the F test.

The smooth transition autoregressive (STAR)(k)¿GARCH(l,¿m) model is a non-linear time series model that is able to account for changes in both regime and volatility respectively. The model can be widely applied to analyse the dynamic behaviour of data exhibiting these two phenomenons in areas such as finance, hydrology and climate change. The main aim of this paper is to perform a Bayesian analysis of STAR(k)¿GARCH(l,¿m) models. The estimation procedure will include estimation of the mean and variance coefficient parameters, the parameters of the transition function, as well as the model orders (k,¿l,¿m). To achieve this aim, the joint posterior distribution of the model orders, coefficient and implicit parameters in the logistic STAR(k)¿GARCH(l,¿m) model is presented. The conditional posterior distributions are then derived, followed by the design of a posterior simulator using a combination of MCMC algorithms which includes Metropolis¿Hastings, Gibbs Sampler and Reversible Jump MCMC algorithms. Following this are extensive simulation studies and a case study presenting the methodology.