Bayesian Nonparametric Predictive Inference and Bootstrap Techniques

1 online resource (PDF, 17 pages

Stark's conjectures

Includes bibliographical references.We give a slightly more general version of the Rubin-Stark conjecture, but show that in most cases it follows from the standard version. After covering the necessary background, we state the principal Stark conjecture and show that although the conjecture depends on a choice of a set of places and a certain isomorphism of Q[GJ-modules, it is independent of these choices. The conjecture is shown to satisfy certain 'functoriality' properties, and we give proofs of the conjecture in some simple cases. The main body of this dissertation concerns a slightly more general version of the Rubin-Stark conjecture. A number of Galois modules. Connected with the conjecture are defined in chapter 4, and some results on exterior powers and Fitting ideals are stated. In chapter 5 the Rubin-Stark conjecture is stated and we show how its truth is unaffected by lowering the top field, changing a set S of places appropriately, and enlarging moduli. We end by giving proofs of the conjecture in several cases. A number of proofs, which would otherwise have interrupted the flow of the exposition, have been relegated to the appendix, resulting in this dissertation suffering from a bad case of appendicitis

THE URYSOHN-MENGER SUM FORMULA: AN EXTENSION OF THE DYDAK-WALSH THEOREM TO DIMENSION ONE

Abstract Let X be a finite-dimensional separable metric space, presented as a disjoint union of subsets, X -A U B. We prove the following theorem: For every prime p, c-dim^^ X < c-din^ A + c-diiri2 p B + 1. This improves upon some of the earlier work by Dydak and Walsh. 1991 Mathematics subject classification (Amer. Math. Soc): primary 55M10, 55P20; secondary 54F45, 55P99

Bayesian Methods for Censored Categorical Data

1 online resource (PDF, 40 pages

Parity-induced Selmer Growth For Symplectic, Ordinary Families

Let $p$ be an odd prime, and let $K/K_0$ be a quadratic extension of number fields. Denote by $K_\pm$ the maximal $\mathbb{Z}_p$-power extensions of $K$ that are Galois over $K_0$, with $K_+$ abelian over $K_0$ and $K_-$ dihedral over $K_0$. In this paper we show that for a Galois representation over $K_0$ satisfying certain hypotheses, if it has odd Selmer rank over $K$ then for one of $K_\pm$ its Selmer rank over $L$ is bounded below by $[L:K]$ for $L$ ranging over the finite subextensions of $K$ in $K_\pm$. Our method or proof generalizes a method of Mazur--Rubin, building upon results of Nekov\'a\v{r}, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families.Comment: 29 page

On aspects of robustness and sensitivity in missing data methods

Missing data are common wherever statistical methods are applied in practice. They present a problem by demanding that additional untestable assumptions be made about the mechanism leading to the incompleteness of the data. Minimising the strength of these assumptions and assessing the sensitivity of conclusions to their possible violation constitute two important aspects of current research in this area. One attractive approach is the doubly robust (DR) weighting-based method proposed by Robins and colleagues. By incorporating two models for the missing data process, inferences are valid when at least one model is correctly specified. The balance between robustness, efficiency and analytical complexity is one which is difficult to strike, resulting in a split between the likelihood and multiple imputation (MI) school on one hand and the weighting and DR school on the other. We propose a new method, doubly robust multiple imputation (DRMI), combining the convenience of MI with the robustness of the DR approach, and explore the use of our new estimator for non-monotone missing at random data, a setting in which, hitherto, estimators with the DR property have not been implemented. We apply the method to data from a clinical trial comparing type II diabetes drugs, where we also use MI as a tool to explore sensitivity to the missing at random assumption. Finally, we study DRMI in the longitudinal binary data setting and find that it compares favourably with existing methods