Special topics in probabilistic exchangeability and its applications

This thesis evolves around a probabilistic concept called exchangeability and its generalised forms. It is aimed at exploring connections between exchangeability and other sub-areas in mathematical statistics. These connections include theoretical implications, generalisation of existing methodologies and applications to real-world data. There are three topics of particular interest. The first topic is related to the linkage between de Finetti's representation theorem (for exchangeable sequences) and existence conditions for Hausdorff moment problems over k-dimensional simplexes. The equivalence of these two results are proved over the most general case in finite spaces. This is a generalisation of existing theory and uses an alternative approach to previous work in the literature. This connection, while theoretically interesting in its own right, may also lead to further cross-field applications, such as distribution re-construction from finite moments or in the approximations to finite exchangeable sequences and finite moment problems. Secondly, we explore a currently popular topic, namely extreme value theory (EVT), which has been widely applied to areas such as hydrology, earth sciences and finance. Classical results from EVT assume that the data sequence is independent and identically distributed (IID). We generalise this assumption to exchangeable random sequences. This caters for more general approaches to EVT that allows for data dependency. Resampling techniques are utilised for estimating the parameters' prior distributions. We utilise these new methods for Value-at-Risk (VaR) estimation in financial stock returns. This is done for both cases with and without GARCH filters. These new VaR models are also compared to existing models in the literature and shows promising improvements. For the final topic, exchangeability is applied to two-phase sampling with an auxiliary variable. In particular, our focus is on a two-phase stratified sampling design, under the assumption that readings for the study variable are exchangeable within stratum. This will again provide a generalisation from the usual IID assumption in applications of multiple-phase sampling. It is amalgamated with stationary bootstrapping at various levels of sampling to estimate within stratum and cross strata covariances. We show that our approach provides a more conservative estimate for the sampling variance of the two-phase estimator for the mean (i.e., the ratio estimator), as compared to the conventional IID method by Rao (1973)