Web access panels are self selected panels constructed with the aim of drawing inference for general populations, including large segments of the population who rarely or never access the Internet. A common approach for modeling survey data collected over access panels is combing it with data collected by a randomly selected reference survey sample from the target population of Interest. The act of joining the panel is then treated as a random process where each member of the population has a positive probability of participating in the survey. The combined reference and panel survey sample can then be used for different estimation approaches which model either the selection process or the measurement of interest, or some case the two together. Most practitioners and academics who have considered this combined sample approach, model the selection process by a single phase process from the target population directly to the observed sample set. In the following work, I assume selection into the panel is a sequential rather than a single phase process and offer several estimators that are underlined by appropriate sequential models. After a careful investigation of a variety of single phase methods applied in practice, I demonstrate the benefits a sequential framework has to the panel problem. One notable strength of this approach is that by assuming a sequential framework the modeler can include important variables associated with Internet and Web usage. Under a single phase model inclusion of such information would invalidate basic assumptions such as independence between selection and model covariates. In this work I also suggest a carefully structured panel estimation strategy, combining a sample selection design with chosen estimator. Under the sequential framework I demonstrate the potential of combining a within-panel random sampling procedure, that is balanced on a sequence of target statistics, with estimators that are modeled over both the selection process and the variable of interest. I show that this strategy has several robustness properties over and beyond currently applied estimators. I conclude by describing an estimation algorithm which applies this estimation strategy to the combined panel and reference survey sample case.

It is a common approach in statistics to assume that the parameters of a stochastic model change. The simplest model involves parameters than can be exactly or approximately piecewise constant. In such a model, the aim is the posteriori detection of the number and location in time of the changes in the parameters. This thesis develops segmentation methods for non-stationary time series and regression models using randomised methods or methods that involve L1 penalties which force the coefficients in a regression model to be exactly zero. Randomised techniques are not commonly found in nonparametric statistics, whereas L1 methods draw heavily from the variable selection literature. Considering these two categories together, apart from other contributions, enables a comparison between them by pointing out strengths and weaknesses. This is achieved by organising the thesis into three main parts. First, we propose a new technique for detecting the number and locations of the change-points in the second-order structure of a time series. The core of the segmentation procedure is the Wild Binary Segmentation method (WBS) of Fryzlewicz (2014), a technique which involves a certain randomised mechanism. The advantage of WBS over the standard Binary Segmentation lies in its localisation feature, thanks to which it works in cases where the spacings between change-points are short. Our main change-point detection statistic is the wavelet periodogram which allows a rigorous estimation of the local autocovariance of a piecewise-stationary process. We provide a proof of consistency and examine the performance of the method on simulated and real data sets. Second, we study the fused lasso estimator which, in its simplest form, deals with the estimation of a piecewise constant function contaminated with Gaussian noise (Friedman et al. (2007)). We show a fast way of implementing the solution path algorithm of Tibshirani and Taylor (2011) and we make a connection between their algorithm and the taut-string method of Davies and Kovac (2001). In addition, a theoretical result and a simulation study indicate that the fused lasso estimator is suboptimal in detecting the location of a change-point. Finally, we propose a method to estimate regression models in which the coefficients vary with respect to some covariate such as time. In particular, we present a path algorithm based on Tibshirani and Taylor (2011) and the fused lasso method of Tibshirani et al. (2005). Thanks to the adaptability of the fused lasso penalty, our proposed method goes beyond the estimation of piecewise constant models to models where the underlying coefficient function can be piecewise linear, quadratic or cubic. Our simulation studies show that in most cases the method outperforms smoothing splines, a common approach in estimating this class of models.

The thesis concerns semiparametric modelling and forecasting Value-at-Risk models, and the applications of these in financial data. Two general classes of semiparametric VaR models are proposed, the first method is introduced by defining some efficient estimators of the risk measures in a semiparametric GARCH model through moment constraints and a quantile estimator based on inverting an empirical likelihood weighted distribution. It is found that the new quantile estimator is uniformly more efficient than the simple empirical quantile and a quantile estimator based on normalized residuals. At the same time, the efficiency gain in error quantile estimation hinges on the efficiency of estimators of the variance parameters. We show that the same conclusion applies to the estimation of conditional Expected Shortfall. The second model proposes a new method to forecast one-period-ahead Value-at-Risk (VaR) in general ARCH(1) models with possibly heavy-tailed errors. The proposed method is based on least square estimation for the log-transformed model. This method imposes weak moment conditions on the errors. The asymptotic distribution also accounts for the parameter uncertainty in volatility estimation. We test our models against some conventional VaR forecasting methods, and the results demonstrate that our models are among the best in forecasting VaR.

The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results.

In this thesis a new method is proposed to estimate major periods of migration from one region into another using phased, non-recombined sequence data from the present. The assumption is made that migration occurs in multiple waves and that during each migration period, a number of sequences, called `founder sequences', migrate into the new region. It is first shown through appropriate simulations based on the structured coalescent that previous inferences based on the idea of founder sequences sufer from the fundamental problem that it is assumed that migration events coincide with the nodes (coalescent events) of the reconstructed tree. It is shown that such an assumption leads to contradictions with the assumed underlying migration process, and that inferences based on such a method have the potential for bias in the date estimates obtained. An improved method is proposed which involves `connected star trees', a tree structure that allows the uncertainty in the time of the migration event to be modelled in a probabilistic manner. Useful theoretical results under this assumption are derived. To model the uncertainty of which founder sequence belongs to which migration period, a Bayesian mixture modelling approach is taken, inferences in which are made by Markov Chain Monte Carlo techniques. Using the developed model, a reanalysis of a dataset that pertains to the settlement of Europe is undertaken. It is shown that sensible inferences can be made under certain conditions using the new model. However, it is also shown that questions of major interest cannot be answered, and certain inferences cannot be made due to an inherent lack of information in any dataset composed of sequences from the present day. It is argued that many of the major questions of interest regarding the migration of modern day humans into Europe cannot be answered without strong prior assumptions being made by the investigator. It is further argued that the same reasons that prohibit certain inferences from being made under the proposed model would remain in any method which has similar assumptions.

What is a latent variable? Simply defined, a latent variable is a variable that cannot be directly measured or observed. A latent variable model or latent structure model is a model whose structure contains one or many latent variables. The subject of this thesis is the study of various topics that arise during the analysis and/or use of latent structure models. Two classical models, namely the factor analysis (FA) model and the finite mixture (FM) model, are first considered and examined extensively, after which the mixture of factor analysers (MFA) model, constructed using ingredients from both FA and FM is introduced and studied at length. Several extensions of the MFA model are also presented, one of which consists of the incorporation of fixed observed covariates into the model. Common to all the models considered are such topics as: (a) model selection which consists of the determination or estimation of the dimensionality of the latent space; (b) parameter estimation which consists of estimating the parameters of the postulated model in order to interpret and characterise the mechanism that produced the observed data; (c) prediction which consists of estimating responses for future unseen observations. Other important topics such as identifiability (for unique solution, interpretability and parameter meaningfulness), density estimation, and to a certain extent aspects of unsupervised learning and exploration of group structure (through clustering, data visualisation in 2D) are also covered. We approach such topics as parameter estimation and model selection from both the likelihood-based and Bayesian perspectives, with a concentration on Maximum Likelihood Estimation via the EM algorithm, and Bayesian Analysis via Stochastic Simulation (derivation of efficient Markov Chain Monte Carlo algorithms). The main emphasis of our work is on the derivation and construction of computationally efficient algorithms that perform well on both synthetic tasks and real-life problems, and that can be used as alternatives to other existing methods wherever appropriate.

In modern society people are concerned about their state of health and if they do unfortunately become ill they want the best possible treatment to be made available to them. In order to satisfy these demands new treatments have to be developed. This can be a long and expensive process. Before any new treatment can come to market it has to be proved to be both clinically effective and economically cost-effective. With limited health care resources the cost-effectiveness of treatments is becoming ever more relevant. In order to show whether a treatment is clinically effective a clinical trial is carried out and this is now usually accompanied by an economic evaluation, so that the cost effectiveness of the treatment can be assessed. When a clinical trial aimed at preventing clinical events is analysed, a time-to-first event analysis is often performed together with a cost-effectiveness analysis. These analyses do not always make the best use of the large amounts of patient information recorded during the clinical trial. Using the randomised controlled trial (RCT) the Impact Of Nicorandil in Angina (IONA) as an exemplar, ways in which the clinical and economic evaluations of clinical trials can be expanded are explored. There are three main parts of this thesis. Firstly, following a more detailed introduction in Chapter 1, in Chapters 2 and 3 the IONA Study is introduced and the main clinical results of the study are given. Secondly, in Chapters 4, 5 and 6 the fact that patients could suffer more than one clinical endpoint is considered. The models that can be used to incorporate the recurrent events are introduced and then applied to the data from the IONA Study. Following on from this, through the simulation of recurrent event data, the performance of the models under different known conditions is assessed. Thirdly, in Chapters 7 and 8 an introduction to health economics is given and following this the main results of the economic evaluation of the IONA Study are presented. Areas in which the results of the economic evaluation can be expanded are then investigated. Finally, in Chapter 9 there is a discussion of the work as a whole and areas where there would be the possibility of further work.

The application of existing geostatistical theory to the context of stream networks provides a number of interesting and challenging problems. The most important of these is how to adapt existing theory to allow for stream, as opposed to Euclidean, distance to be used. Valid stream distance based models for the covariance structure have been denied in the literature, and this thesis explores the use of such models using data from the River Tweed. The data span a period of twenty-one years, beginning in 1986. During this time period, up to eighty-three stations are monitored for a variety of chemical and biological determinands. This thesis will focus on nitrogen, a key nutrient in determining water quality, especially given the Nitrates Directive (adopted in 1991) and the Water Framework Directive(adopted in 2002). These are European Union legislations that have set legally enforcable guidelines for controlling pollution which national bodies must comply with. The focus of analysis is on several choices that must be made in order to carry out spatial prediction on a river network. The role of spatial trend, whether it be based on stream or Euclidean distance, is discussed and the impact of the bandwidth of the estimate of nonparametric trend is explored. The stream distance based "tail-up" covariance model structure of Ver Hoef and Peterson (2010) is assessed and combined with a standard Euclidean distance based structure to form a mixture model. This is then evaluated using crossvalidation studies in order to determine the optimum mixture of the two covariance models for the data. Finally, the covariance models used for each of the elements of the mixture model are explored to determine the impact they have on the lowest root mean squared error, and the mixing proportion at which it is found. Using the predicted values at unobserved locations on the River Tweed, the distribution of yearly averaged nitrate levels around the river network is predicted and evaluated. Changes through the 21 years of data are noted and areas exceeding the limits set by the Nitrates Directive are highlighted. The differences in fitted values caused by using stream or Euclidean distance are evident in these predictions. The data is then modelled through space and time using additive models. A novel smoothing function for the spatial trend is defined. It is adapted from the tail-up model in order to retain its core features of flow connectivity and flow volume based weightings, in addition to being based on stream distance. This is then used to model all of the River Tweed data through space and time and identify temporal trends and seasonal patterns at different locations on the river.

Longitudinal data requires special statistical methods because the observations on one subject tend to be correlated. (Although subjects can usually be assumed to be independent). When subjects are individually observed at varying sets of times with or without missing data, as is the case of ALSPAC data during pregnancy, then the resulting data is referred to as unbalanced data. This can cause further complications for the analysis. The aim of this thesis is to contribute to longitudinal research of this topic by using mixed-effects models, which provide a powerful and flexible tool for the analysis of balanced and unbalanced data. Although progress has been made in the study reported in this thesis, further extensions are required. As the longitudinal data typically need some structured covariance models, the overall findings indicate that when the number of occasions is large with some missing values, the use of polynomial function is inadequate to describe the model. This study highlights an approach that applies cubic spline in longitudinal modelling, including an emphasis on the use of graphical representation for exploratory analysis and the assessment of model fit. Cubic splines provide a flexible tool for longitudinal data. The main objective of this study is to investigate a methodology to incorporate cubic spline with linear mixed models in modelling longitudinal data with number of time points and missing values.

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