An essential aspect of survival analysis is the estimation and prediction of survival probabilities for individuals. For this purpose, mathematical modelling of the hazard rate function is a fundamental issue. This thesis focuses on the novel estimation and application of hazard rate functions in mathematical and medical research. In mathematical research we focus on the development of a semiparametric kernel-based estimate of hazard rate function and a L\(_1\) error optimal kernel hazard rate estimate. In medical research we concentrate on the development and validation of survival models using individual participant data from multiple studies. We also consider how to fit survival models that predict individual response to treatment effectiveness, given IPD from multiple trials.

Empirical testing is centred on p-values. These summary statistics are used to assess the plausibility of a null hypothesis, and therein lies a flaw in their interpretation. Central to this research is accounting for the behaviour of p-values, through density functions, under the alternative hypothesis, H1. These densities are determined by a combination of the sample size and parametric specification of H1. Here, several new contributions are presented to reflect p-value behaviour. By considering the likelihood of both hypotheses in parallel, it is possible to optimise the decision-making process. A framework for simultaneously testing the null and alternative hypotheses is outlined for various testing scenarios. To facilitate efficient empirical conclusions, a new set of critical value tables is presented requiring only the conventional p-value, hence avoiding the need for additional computation in order to apply this joint testing in practice. Simple and composite forms of H1 are considered. Recognising the conflict between different schools of thought with respect to hypothesis testing, a unified approach at consolidating the advantages of each is offered. Again, exploiting p-value distributions under various forms of H1, a revised conditioning statistic for conditional frequentist testing is developed from which original p-value curves and surfaces are produced to further ease decision making. Finally, attention turns to multiple hypothesis testing. Estimation of multiple testing error rates is discussed and a new estimator for the proportion of true null hypotheses, when simultaneously testing several independent hypotheses, is presented. Under certain conditions it is shown that this estimator is superior to an established estimator.

A decision maker, when facing a decision problem, often considers several models to represent the outcomes of the decision variable considered. More often than not, the decision maker does not trust fully any of those models and hence displays ambiguity or model uncertainty aversion. In this PhD thesis, focus is given to the specific case of asset allocation problem under ambiguity faced by financial investors. The aim is not to find an optimal solution for the investor, but rather come up with a general methodology that can be applied in particular to the asset allocation problem and allows the investor to find a tractable, easy to compute solution for this problem, taking into account ambiguity. This PhD thesis is structured as follows: First, some classical and widely used models to represent asset returns are presented. It is shown that the performance of the asset portfolios built using those single models is very volatile. No model performs better than the others consistently over the period considered, which gives empirical evidence that: no model can be fully trusted over the long run and that several models are needed to achieve the best asset allocation possible. Therefore, the classical portfolio theory must be adapted to take into account ambiguity or model uncertainty. Many authors have in an early stage attempted to include ambiguity aversion in the asset allocation problem. A review of the literature is studied to outline the main models proposed. However, those models often lack flexibility and tractability. The search for an optimal solution to the asset allocation problem when considering ambiguity aversion is often difficult to apply in practice on large dimension problems, as the ones faced by modern financial investors. This constitutes the motivation to put forward a novel methodology easily applicable, robust, flexible and tractable. The Ambiguity Robust Adjustment (ARA) methodology is theoretically presented and then tested on a large empirical data set. Several forms of the ARA are considered and tested. Empirical evidence demonstrates that the ARA methodology improves portfolio performances greatly. Through the specific illustration of the asset allocation problem in finance, this PhD thesis proposes a new general methodology that will hopefully help decision makers to solve numerous different problems under ambiguity.

Sparse modelling has attracted great attention as an efficient way of handling statistical problems in high dimensions. This thesis considers sparse modelling and estimation in a selection of problems such as breakpoint detection in nonstationary time series, nonparametric regression using piecewise constant functions and variable selection in high-dimensional linear regression. We first propose a method for detecting breakpoints in the secondorder structure of piecewise stationary time series, assuming that those structural breakpoints are sufficiently scattered over time. Our choice of time series model is the locally stationary wavelet process (Nason et al., 2000), under which the entire second-order structure of a time series is described by wavelet-based local periodogram sequences. As the initial stage of breakpoint detection, we apply a binary segmentation procedure to wavelet periodogram sequences at each scale separately, which is followed by within-scale and across-scales postprocessing steps. We show that the combined methodology achieves consistent estimation of the breakpoints in terms of their total number and locations, and investigate its practical performance using both simulated and real data. Next, we study the problem of nonparametric regression by means of piecewise constant functions, which are known to be flexible in approximating a wide range of function spaces. Among many approaches developed for this purpose, we focus on comparing two well-performing techniques, the taut string (Davies & Kovac, 2001) and the Unbalanced Haar (Fryzlewicz, 2007) methods. While the multiscale nature of the latter is easily observed, it is not so obvious that the former can also be interpreted as multiscale. We provide a unified, multiscale representation for both methods, which offers an insight into the relationship between them as well as suggesting some lessons that both methods can learn from each other. Lastly, one of the most widely-studied applications of sparse modelling and estimation is considered, variable selection in high-dimensional linear regression. High dimensionality of the data brings in many complications including (possibly spurious) non-negligible correlations among the variables, which may result in marginal correlation being unreliable as a measure of association between the variables and the response. We propose a new way of measuring the contribution of each variable to the response, which adaptively takes into account high correlations among the variables. A key ingredient of the proposed tilting procedure is hard-thresholding sample correlation of the design matrix, which enables a data-driven switch between the use of marginal correlation and tilted correlation for each variable. We study the conditions under which this measure can discriminate between relevant and irrelevant variables, and thus be used as a tool for variable selection. In order to exploit these theoretical properties of tilted correlation, we construct an iterative variable screening algorithm and examine its practical performance in a comparative simulation study.

The purpose of this research was to use Bayesian statistics to develop flexible mortality models that could be used to forecast human mortality rates. Several models were developed as extensions to existing mortality models, in particular the Lee-Carter mortality model and the age-period-cohort model, by including some of the following features: age-period and age-cohort interactions, random effects on mortality, measurement errors in population count and smoothing of the mortality rate surface. One expects mortality rates to change in a relatively smooth manner between neighbouring ages or between neighbouring years or neighbouring cohorts. The inclusion of random effects in some of the models captures additional fluctuations in these effects. This smoothing is incorporated in the models by ensuring that the age, period and cohort parameters of the models have a relatively smooth sequence which is achieved through the choice of the prior distribution of the parameters. Three different smoothing priors were employed: a random walk, a random walk on first differences of the parameters and an autoregressive model of order one on the first differences of the parameters. In any model only one form of smoothing was used. The choice of smoothing prior not only imposes different patterns of smoothing on the parameters but is seen to be very influential when making mortality forecasts. The mortality models were fitted, using Bayesian methods, to population data for males and females from England and Wales. The fits of the models were analysed and compared using analysis of residuals, posterior predictive intervals for both in-sample and out-of-sample data and the Deviance Information Criterion. The models fitted the data better than did both the Lee-Carter model and the age-period-cohort model. From the analysis undertaken, for any given age and calendar year, the preferred model based on the Deviance Information Criterion score, for male and female death counts was a Poisson model with the mean parameter equal to the number of lives exposed to risk of dying for that age in that calendar year multiplied by a mortality parameter. The logit of this mortality parameter was a function of the age, year (period) and cohort with additional interactions between the age and period parameters and between the age and cohort parameters. The form of parameter smoothing that suited the males was an autoregressive model of order one on the first differences of the parameters and that for the females was a random walk. Moreover, it was found useful to add Gaussian random effects to account for overdispersion caused by unobserved heterogeneity in the population mortality. The research concluded by the application of a selection of these models to the provision of forecasts of period and cohort life expectancies as well as the numbers of centenarians for males and females in England and Wales. In addition, the thesis illustrated how Bayesian mortality models could be used to consider the impact of the new European Union solvency regulations for insurers (Solvency II) for longevity risk. This research underlined the important role that Bayesian stochastic mortality models can play in considering longevity risk.

A covariance matrix of asset returns plays an important role in modern portfolio analysis and risk management. Despite the recent interests in improving the estimation of a return covariance matrix, there remain many areas for further investigation. This thesis studies several issues related to obtaining a better estimation of the covariance matrix for the returns of a reasonably large number of stocks for portfolio risk management. The thesis consists of five essays. The first essay, Chapter 3, provides a comprehensive analysis of both old and new covariance estimation methods and the standard comparison criteria. We use empirical data to compare their performances. We also examine the standard comparisons and find they provide limited information regarding the abilities of the covariance estimators in predicting portfolio variances. It therefore suggests that we need more powerful comparison criteria to assess covariance estimators. The second and third essays, Chapter 4 and 5, are concerned with the alternative appraisal methods of return covariance estimators for portfolio risk management purposes. Chapter 4 introduces a portfolio distance measure based on eigen decomposition (eigen-distance) to compare two covariance estimators in terms of the most different portfolio variances they predict. The eigen-distance measures the ratio of the two extreme variance predictions under one covariance estimator for the portfolios that are constructed to have the same variances under the other covariance estimator. We show that the eigen-distance can be used to assess a risk measurement system as a whole, where any kind of the portfolios may need to be considered. Our simulation results show that it is a powerful measure to distinguish two covariance estimators even in small samples. Chapter 5 proposes a0 measure to distinguish two similar estimated covariance matrices from the observed covariance matrix. 0 is constructed based on the essential difference of the two similar covariance matrices: the two extreme portfolios that are predicted to have the most different variances under these two matrices. We show that 0 is very useful in evaluating refinements to covariance estimators, particularly a modest refinement, where the refined covariance matrix is close to the original matrix. The last two essays, Chapter 6 and 7, are concerned with improving the best covariance estimators within the literature. Chapter 6 explores alternative Bayesian shrinkage methods that directly shrink the eigenvalues (and in one case the principal eigenvector) of the sample covariance matrix. We use simulations to compare the performance of these shrinkage estimators with the two best existing estimators, namely, the Ledoit and Wolf (2003a) estimator and the Jagannathan and Ma (2003) estimator using both RMSE and eigen-distance criteria. We find that our shrinkage estimators consistently out-perform the Ledoit and Wolf estimator. They also out-perform the Jagannathan and Ma estimator except in one case where they are not much worse off either. Finally, Chapter 7 extends the analysis of Chapter 6, which is under an unchanging multivariate normal world, to consider implications of both fat-tails and time variation. We use a multivariate normal inverse Gaussian (MNIG) distribution to model the log returns of stock prices. This family of distributions has proven to fit the heavy tails observed in financial time series extremely well. For the time varying situation, we use a tractable mean reverting Ornstein- Uhlenbeck (OU) process to develop a new model to measure an interesting and economically motivated time varying structure where the risks remain unchanged but stocks migrate among different risk categories during their life circles. We find that our shrinkage methods are also useful in both situations and become even more important in the time varying case.

Multiple imputation is a commonly used approach to deal with missing data and to protect confidentiality of public use data sets. The basic idea is to replace the missing values or sensitive values with multiple imputation, and we then release the multiply imputed data sets to the public. Users can analyze the multiply imputed data sets and obtain valid inferences by using simple combining rules, which take the uncertainty due to the presence of missing values and synthetic values into account. It is crucial that imputations are drawn from the posterior predictive distribution to preserve relationships present in the data and allow valid conclusions to be made from any analysis. In data sets with different types of variables, e.g. some categorical and some continuous variables, multivariate imputation by chained equations (MICE) (Van Buuren (2011)) is a commonly used multiple imputation method. However, imputations from such an approach are not necessarily drawn from a proper posterior predictive distribution. We propose a method, called factored regression model (FRM) to multiply impute missing values in such data sets by modelling the joint distribution of the variables in the data through a sequence of generalised linear models. We use data augmentation methods to connect the categorical and continuous variables and this allows us to draw imputations from a proper posterior distribution. We compare the performance of our method with MICE using simulation studies and on a breastfeeding data. We also extend our modelling strategies to incorporate different informative priors for the FRM to explore robust regression modelling and the sparse relationships between the predictors. We then apply our model to protect confidentiality of the current population survey (CPS) data by generating multiply imputed, partially synthetic data sets. These data sets comprise a mix of original data and the synthetic data where values chosen for synthesis are based on an approach that considers unique and sensitive units in the survey. Valid inference can then be made using the combining rules described by Reiter (2003). An extension to the modelling strategy is also introduced to deal with the presence of spikes at zero in some of the continuous variables in the CPS data.

The thesis herein explores the relationship between early and late information in judgement and decision-making and tests a quantitative model of this relationship based on contemporary dual-process theory. The first chapter reviews literature regarding early information as a potential biasing factor in judgement and decision-making, the neglect of dual-process theory in the domain and the tendency to rely on static modelling techniques derived from economic theory. The first empirical chapter concludes that a synthesis of a static-economic decision model (prospect theory) with contemporary dual-process theory principles can better predict choice behaviour than either one approach alone. I conclude that dual-process theory provides a strong theoretical basis for understanding the cognitive processes involved in early-informational biases, but also that the quantitative approaches to modelling choice behaviour can provide valuable additional insights. The third chapter acts on this conclusion by developing a dynamic-stochastic choice model (based on a sequential sample process) which reflects four contemporary dual-process theory concepts that are relevant to early-informational biases. Simulation results of the model are presented in order to demonstrate the choice behaviour predicted by this approach. The rest of the thesis is dedicated to empirical studies designed to test the implications of these simulation results and these predicted behaviours. The empirical studies cover a range of domains including biased predecision processing during evidence gathering, stereotype bias in multi-attribute decision-making under time-pressure and the impact of expectation and accuracy motivation on visual-search decision-making. I conclude that the dynamic-stochastic modelling approach demonstrates some clear value in understanding the cognitive processes involved in these domains and the results support the use of contemporary dual-process theory as a framework for understanding judgement and decision-making. Based on this conclusion I outline some future developments for a more nuanced dynamic model including integration with a more sophisticated way of modelling type 2 processing and expansion to account for hypothetical thinking principles. I also suggest future research domains for application of the model such as expert decision-making and multi-alternative decision problems.

This thesis focuses on how it is possible to measure people’s knowledge on a topic where certain statements can effectively discriminate between knowledgeable and non knowledgeable people. It presents an application in measuring healthcare professionals’ knowledge about Sudden Infant Death Syndrome (SIDS) and its risk factors. Identifying the best and worst prepared healthcare professionals allows policymakers to reconsider the structure of their healthcare system and to implement targeted training initiatives about this topic. To do so, this research uses data belonging to the SIDS Project, a project meant to provide the first data about this topic in the United Kingdom and Spain. The mail survey referring to the United Kingdom was carried out in the South Central Strategic Health Authority in 2012, while the Spanish one was carried out in the provinces of Barcelona, Lérida and Tarragona in 2012 and 2013. The target population for the British survey consisted of general practitioners (GPs), while the target population for the Spanish survey consisted of paediatricians. Moreover, data about Italy were also available, which allowed cross country comparisons involving three different realities. This research shows that the Back-To-Sleep (BTS) message seems to have been effectively adopted by the British GPs, but, surprisingly, not as well received by the Spanish and Italian paediatricians. In the first case, in fact, more than 90% of the respondents recommended parents the supine position exclusively. In Spain and Italy, instead, this percentage was of 58% and 69% respectively. By contract, instead, the whole SIDS prevention message seems to have been better received in Spain and Italy than in the United Kingdom. British policymakers should reconsider the role of GPs in terms of delivering parents the BTS message, as they were found to be quite prepared. Spanish and Italian policymakers, instead, should try to increase the degree of adoption of the BTS message among their healthcare professionals. In particular, Spanish policymakers should urgently intervene in order to clarify that the supine position is the only one that can be deemed to be a protective factor against SIDS.

A cross-over study is a comparitive experiment in which subjects receive a sequence of two or more treatments, one in each of a series of successive time periods, and the response of each subject is measured at the end of every period. A common problem, particularly in medicine, is that subjects fail to complete a study through dropping out during the later stages of the trial for reasons unrelated to the treatments received. Current practice is to select a design for a study on the basis of its performance under the assumption that no subjects drop out, using a criterion such as A-optimality. This is an unrealistic assumption for many medical applications. This thesis investigates how studies should be designed when it is unrealistic to assume that subjects will not drop out. A method of assessing cross-over designs is presented which judges how accurately all the pairwise treatment comparisons are estimated under the assumption that each subject has a fixed probability of dropping out during the final period, independent of treatment received and the other subjects. The method of design assessment is computationally intensive even for studies involving a relatively small number of subjects. Ways of reducing the amount of computation required are presented through establishing the link between implemented designs and a colouring problem in combinatorial theory. The reductions achieved make feasible investigations of currently used designs for cross-over studies. The results of investigations are presented for designs for the cases of particular practical importance, namely four treatment, four period and three treatment, three period studies, in which a simple carry-over model is assumed for the observations. Designs which are more robust to final period dropout than the currently favoured designs are identified.