This dissertation explores the possibility of using an imaging approach to model classical pharmacokinetic (PK) problems. The kinetic parameters which describe the uptake rates of a drug within a biological system, are parameters of interest. Knowledge of the drug uptake in a system is useful in expediting the drug development process, as well as providing a dosage regimen for patients. Traditionally, the uptake rate of a drug in a system is obtained via sampling the concentration of the drug in a central compartment, usually the blood, and fitting the data to a curve. In a system consisting of multiple compartments, the number of kinetic parameters is proportional to the number of compartments, and in classical PK experiments, the number of identifiable parameters is less than the total number of parameters. Using an imaging approach to model classical PK problems, the support region of each compartment within the system will be exactly known, and all the kinetic parameters are uniquely identifiable. To solve for the kinetic parameters, an indirect approach, which is a two part process, was used. First the compartmental activity was obtained from data, and next the kinetic parameters were estimated. The novel aspect of the research is using listmode data to obtain the activity curves from a system as opposed to a traditional binned approach. Using techniques from information theoretic learning, particularly kernel density estimation, a non-parametric probability density function for the voltage outputs on each photo-multiplier tube, for each event, was generated on the fly, which was used in a least squares optimization routine to estimate the compartmental activity. The estimability of the activity curves for varying noise levels as well as time sample densities were explored. Once an estimate for the activity was obtained, the kinetic parameters were obtained using multiple cost functions, and the compared to each other using the mean squared error as the figure of merit.

Statistics is a branch of mathematics which is heavily employed in the area of Actuarial Mathematics. This thesis first reviews the importance of statistical distributions in the analysis of insurance problems and the applications of Statistics in the area of risk and insurance. The Normal, Log-normal, Pareto, Gamma, standard Beta, Frechet, Gumbel, Weibull, Poisson, binomial, and negative binomial distributions are looked at and the importance of these distributions in general insurance is also emphasized. A careful review of literature is to provide practitioners in the general insurance industry with statistical tools which are of immediate application in the industry. These tools include estimation methods and fit statistics popular in the insurance industry. Finally this thesis carries out the task of fitting statistical distributions to the flood loss data in the 50 States of the United States.

Definimos, a partir de uma partição de um intervalo limitado da reta real formada por subintervalos, uma distribuição a priori sobre uma classe de densidades em relação à medida de Lebesgue construindo uma densidade aleatória cujas realizações são funções simples não negativas que assumem um valor constante em cada subintervalo da partição e possuem integral unitária. Utilizamos tais densidades aleatórias simples na análise bayesiana de um conjunto de observáveis absolutamente contínuos e provamos que a distribuição a priori é fechada sob amostragem. Exploramos as distribuições a priori e a posteriori via simulações estocásticas e obtemos soluções bayesianas para o problema de estimação de densidade. Os resultados das simulações exibem o comportamento assintótico da distribuição a posteriori quando crescemos o tamanho das amostras dos dados analisados. Quando a partição não é conhecida a priori, propomos um critério de escolha a partir da informação contida na amostra. Apesar de a esperança de uma densidade aleatória simples ser sempre uma densidade descontínua, obtemos estimativas suaves resolvendo um problema de decisão em que os estados da natureza são realizações da densidade aleatória simples e as ações são densidades suaves de uma classe adequada. / We define, from a known partition in subintervals of a bounded interval of the real line, a prior distribution over a class of densities with respect to Lebesgue measure constructing a random density whose realizations are nonnegative simple functions that integrate to one and have a constant value on each subinterval of the partition. These simple random densities are used in the Bayesian analysis of a set of absolutely continuous observables and the prior distribution is proved to be closed under sampling. We explore the prior and posterior distributions through stochastic simulations and find Bayesian solutions to the problem of density estimation. Simulations results show the asymptotic behavior of the posterior distribution as we increase the size of the analyzed data samples. When the partition is unknown, we propose a choice criterion based on the information contained in the sample. In spite of the fact that the expectation of a simple random density is always a discontinuous density, we get smooth estimates solving a decision problem where the states of nature are realizations of the simple random density and the actions are smooth densities of a suitable class.

Definimos, a partir de uma partição de um intervalo limitado da reta real formada por subintervalos, uma distribuição a priori sobre uma classe de densidades em relação à medida de Lebesgue construindo uma densidade aleatória cujas realizações são funções simples não negativas que assumem um valor constante em cada subintervalo da partição e possuem integral unitária. Utilizamos tais densidades aleatórias simples na análise bayesiana de um conjunto de observáveis absolutamente contínuos e provamos que a distribuição a priori é fechada sob amostragem. Exploramos as distribuições a priori e a posteriori via simulações estocásticas e obtemos soluções bayesianas para o problema de estimação de densidade. Os resultados das simulações exibem o comportamento assintótico da distribuição a posteriori quando crescemos o tamanho das amostras dos dados analisados. Quando a partição não é conhecida a priori, propomos um critério de escolha a partir da informação contida na amostra. Apesar de a esperança de uma densidade aleatória simples ser sempre uma densidade descontínua, obtemos estimativas suaves resolvendo um problema de decisão em que os estados da natureza são realizações da densidade aleatória simples e as ações são densidades suaves de uma classe adequada. / We define, from a known partition in subintervals of a bounded interval of the real line, a prior distribution over a class of densities with respect to Lebesgue measure constructing a random density whose realizations are nonnegative simple functions that integrate to one and have a constant value on each subinterval of the partition. These simple random densities are used in the Bayesian analysis of a set of absolutely continuous observables and the prior distribution is proved to be closed under sampling. We explore the prior and posterior distributions through stochastic simulations and find Bayesian solutions to the problem of density estimation. Simulations results show the asymptotic behavior of the posterior distribution as we increase the size of the analyzed data samples. When the partition is unknown, we propose a choice criterion based on the information contained in the sample. In spite of the fact that the expectation of a simple random density is always a discontinuous density, we get smooth estimates solving a decision problem where the states of nature are realizations of the simple random density and the actions are smooth densities of a suitable class.

Ensemble techniques have recently been used to enhance the performance of machine learning methods. However, current ensemble techniques for classification require both positive and negative data to produce a result that is both meaningful and useful. Negative data is, however, sometimes difficult, expensive or impossible to access. In this thesis a learning framework is described that has a very close relationship to boosting. Within this framework a method is described which bears remarkable similarities to boosting stumps and that does not rely on negative examples. This is surprising since learning from positive-only data has traditionally been difficult. An empirical methodology is described and deployed for testing positive-only learning systems using commonly available multiclass datasets to compare these learning systems with each other and with multiclass learning systems. Empirical results show that our positive-only boosting-like method learns, using stumps as a base learner and from positive data only, successfully, and in the process does not pay too heavy a price in accuracy compared to learners that have access to both positive and negative data. We also describe methods of using positive-only learners on multiclass learning tasks and vice versa and empirically demonstrate the superiority of our method of learning in a boosting-like fashion from positive-only data over a traditional multiclass learner converted to learn from positive-only data. Finally we examine some alternative frameworks, such as when additional unlabelled training examples are given. Some theoretical justifications of the results and methods are also provided.

Sigmoid type belief networks, a class of probabilistic neural networks, provide a natural framework for compactly representing probabilistic information in a variety of unsupervised and supervised learning problems. Often the parameters used in these networks need to be learned from examples. Unfortunately, estimating the parameters via exact probabilistic calculations (i.e, the EM-algorithm) is intractable even for networks with fairly small numbers of hidden units. We propose to avoid the infeasibility of the E step by bounding likelihoods instead of computing them exactly. We introduce extended and complementary representations for these networks and show that the estimation of the network parameters can be made fast (reduced to quadratic optimization) by performing the estimation in either of the alternative domains. The complementary networks can be used for continuous density estimation as well.

The k-sample problem, i.e., testing whether two or more data sets come from the same population, is a classic one in statistics. Instead of having a small number of k groups of samples, this dissertation works on a large number of p groups of samples, where within each group, the sample size, n, is a fixed, small number. We call this as a "Large p, but Small n" setting. The primary goal of the research is to provide a test statistic based on kernel density estimation (KDE) that has an asymptotic normal distribution when p goes to infinity with n fixed. In this dissertation, we propose a test statistic called Tp(S) and its standardized version, T(S). By using T(S), we conduct our test based on the critical values of the standard normal distribution. Theoretically, we show that our test is invariant to a location and scale transformation of the data. We also find conditions under which our test is consistent. Simulation studies show that our test has good power against a variety of alternatives. The real data analyses show that our test finds differences between gene distributions that are not due simply to location.

This thesis, which consists of four chapters, focuses on three topics: discriminating between stationary and nonstationary time series, testing the constancy of the error covariance matrix of a vector model, and estimating density functions over bounded domains using kernel techniques. In Chapter 1, “Testing the unit root hypothesis against the logistic smooth transition autoregressive model”, and Chapter 2, “A nonlinear alternative to the unit root hypothesis”, the joint hypothesis of unit root and linearity allows one to distinguish between random walk processes, with or without drift, and stationary nonlinear processes of the smooth transition autoregressive type. This is important in applications because steps taken in modelling a time series are likely to be drastically different depending on whether or not the unit root hypothesis is rejected. In Chapter 1 the nonlinearity is based on the logistic function, while Chapter 2 considers the second-order logistic function. Monte Carlo simulations show that the proposed tests have about the same or higher power than the standard Dickey-Fuller unit root tests when the alternative exhibits nonlinear behavior. In Chapter 1 the tests are applied to the seasonally adjusted U.S. monthly unemployment rate, giving support to the hypothesis that the unemployment rate series follows a smooth transition autoregressive model rather than a random walk. Chapter 2 considers testing the so called purchasing power parity (PPP) hypothesis. The test results complement earlier studies, giving support to the PPP hypothesis for 44 out of 120 real exchange rates considered. Chapter 3. “Testing the constancy of the error covariance matrix in vector models”Estimating the parameters of an econometric model is necessary for any use of the model, be it forecasting or policy evaluation. Finding out thereafter whether or not the model appears to satisfy the assumptions under which it was estimated should be an integral part of a normal modelling exercise. This chapter includes the derivation of a Lagrange Multiplier test of the null hypothesis of constant variance in vector models when testing against three specific parametric alternatives. The Monte Carlo simulations show that the test has good size properties, very good power against a correctly specified alternative, but low or only up to moderate power in cases for a misspecified alternative hypothesis. Chapter 4. “ Estimating confidence regions over bounded domains”Nonparametric density estimation by kernel techniques is a standard statistical tool in the estimation of a density function in situations where its parametric form is assumed to be unknown. In many cases, the data set over which the density is to be estimated exhibits linear, or nonlinear, dependence. A solution to this problem is to apply a one-to-one transformation to the considered data set in such a way that the dependence in the data vanishes, but too often such a unique transformation does not exist. This chapter proposes a method for estimating confidence regions over bounded domains when no one-to-one transformation of the considered data exists, or if the existence of such a transformation is difficult to verify. The method, simple kernel estimation over a nonlinear grid, is illustrated by applying it to three data sets generated from the GARCH(1,1) model. The resulting confidence regions cover a reasonable area of the definition space, and are well aligned with the corresponding data sets. / Diss. Stockholm : Handelshögsk., 2003

This thesis develops models and associated Bayesian inference methods for flexible univariate and multivariate conditional density estimation. The models are flexible in the sense that they can capture widely differing shapes of the data. The estimation methods are specifically designed to achieve flexibility while still avoiding overfitting. The models are flexible both for a given covariate value, but also across covariate space. A key contribution of this thesis is that it provides general approaches of density estimation with highly efficient Markov chain Monte Carlo methods. The methods are illustrated on several challenging non-linear and non-normal datasets. In the first paper, a general model is proposed for flexibly estimating the density of a continuous response variable conditional on a possibly high-dimensional set of covariates. The model is a finite mixture of asymmetric student-t densities with covariate-dependent mixture weights. The four parameters of the components, the mean, degrees of freedom, scale and skewness, are all modeled as functions of the covariates. The second paper explores how well a smooth mixture of symmetric components can capture skewed data. Simulations and applications on real data show that including covariate-dependent skewness in the components can lead to substantially improved performance on skewed data, often using a much smaller number of components. We also introduce smooth mixtures of gamma and log-normal components to model positively-valued response variables. In the third paper we propose a multivariate Gaussian surface regression model that combines both additive splines and interactive splines, and a highly efficient MCMC algorithm that updates all the multi-dimensional knot locations jointly. We use shrinkage priors to avoid overfitting with different estimated shrinkage factors for the additive and surface part of the model, and also different shrinkage parameters for the different response variables. In the last paper we present a general Bayesian approach for directly modeling dependencies between variables as function of explanatory variables in a flexible copula context. In particular, the Joe-Clayton copula is extended to have covariate-dependent tail dependence and correlations. Posterior inference is carried out using a novel and efficient simulation method. The appendix of the thesis documents the computational implementation details. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: In press. Paper 4: Manuscript.</p>

Being able to explore a relationship between two life events is of great interest to scientists from different disciplines. Some issues of particular concern are, for example, the connection between smoking cessation and pregnancy (Thompson and Pantoja-Galicia 2003), the interrelation between entry into marriage for individuals in a consensual union and first pregnancy (Blossfeld and Mills 2003), and the association between job loss and divorce (Charles and Stephens 2004, Huang 2003 and Yeung and Hofferth 1998). Establishing causation in observational studies is seldom possible. Nevertheless, if one of two events tends to precede the other closely in time, a causal interpretation of an association between these events can be more plausible. The role of longitudinal surveys is crucial, then, since they allow sequences of events for individuals to be observed. Thompson and Pantoja-Galicia (2003) discuss in this context several notions of temporal association and ordering, and propose an approach to investigate a possible relationship between two lifetime events. In longitudinal surveys individuals might be asked questions of particular interest about two specific lifetime events. Therefore the joint distribution might be advantageous for answering questions of particular importance. In follow-up studies, however, it is possible that interval censored data may arise due to several reasons. For example, actual dates of events might not have been recorded, or are missing, for a subset of (or all) the sampled population, and can be established only to within specified intervals. Along with the notions of temporal association and ordering, Thompson and Pantoja-Galicia (2003) also discuss the concept of one type of event "triggering" another. In addition they outline the construction of tests for these temporal relationships. The aim of this thesis is to implement some of these notions using interval censored data from longitudinal complex surveys. Therefore, we present some proposed tools that may be used for this purpose. This dissertation is divided in five chapters, the first chapter presents a notion of a temporal relationship along with a formal nonparametric test. The mechanisms of right censoring, interval censoring and left truncation are also overviewed. Issues on complex surveys designs are discussed at the end of this chapter. For the remaining chapters of the thesis, we note that the corresponding formal nonparametric test requires estimation of a joint density, therefore in the second chapter a nonparametric approach for bivariate density estimation with interval censored survey data is provided. The third chapter is devoted to model shorter term triggering using complex survey bivariate data. The semiparametric models in Chapter 3 consider both noncensoring and interval censoring situations. The fourth chapter presents some applications using data from the National Population Health Survey and the Survey of Labour and Income Dynamics from Statistics Canada. An overall discussion is included in the fifth chapter and topics for future research are also addressed in this last chapter.