KERNEL REGRESSION SUBJECT TO INTERVAL-CENSORED RESPONSES AND QUALITATIVE CONSTRAINTS

This thesis is concerned with problems arising when one wants to apply flexible non- parametric local regression models to data when there is additional qualitative in­ formation. It is also concerned with nonparametric regression problems involving interval-censored responses. These problems are studied via asymptotic theory where possible and by simulation. Iterated conditional expectation methods and local likelihood estimation for nonparametric interval-censored regression are developed. Simulation results show that local likelihood estimation is often superior to local regression estimators when observations have been imputed using either interval midpoints or iterated conditional expectations when the censoring intervals are wide or of varying width. When the intervals are smaller and of fixed width, none of the imputation approaches dominate the others. Constrained data sharpening for nonparametric regression is applied to new situations such as where constraints are defined by convexity, concavity, and in terms of differential operators. Data sharpening is compared with competing kernel methods in terms of bias, variance and MISE. It is proved that the constrained data sharpening estimator has the same rate of convergence as the constrained weighting estimator of Hall and Huang (2001). Also, penalized data sharpening is proposed as a new form of constrained data sharpening. The sharpened responses can be computed analytically which makes the method very convenient, both for studying theoretically and for applying practicall

Isotonic Distributional Regression

Distributional regression estimates the probability distribution of a response variable conditional on covariates. The estimated conditional distribution comprehensively summarizes the available information on the response variable, and allows to derive all statistical quantities of interest, such as the conditional mean, threshold exceedance probabilities, or quantiles. This thesis develops isotonic distributional regression, a method for estimating conditional distributions under the assumption of a monotone relationship between covariates and a response variable. The response variable is univariate and real-valued, and the covariates lie in a partially ordered set. The monotone relationship is formulated in terms of stochastic order constraints, that is, the response variable increases in a stochastic sense as the covariates increase in the partial order. This assumption alone yields a shape-constrained non-parametric estimator, which does not involve any tuning parameters. The estimation of distributions under stochastic order restrictions has already been studied for various stochastic orders, but so far only with totally ordered covariates. Apart from considering more general partially ordered covariates, the first main contribution of this thesis lies in a shift of focus from estimation to prediction. Distributional regression is the backbone of probabilistic forecasting, which aims at quantifying the uncertainty about a future quantity of interest comprehensively in the form of probability distributions. When analyzed with respect to predominant criteria for probabilistic forecast quality, isotonic distributional regression is shown to have desirable properties. In addition, this thesis develops an efficient algorithm for the computation of isotonic distributional regression, and proposes an estimator under a weaker, previously not thoroughly studied stochastic order constraint. A main application of isotonic distributional regression is the uncertainty quantification for point forecasts. Such point forecasts sometimes stem from external sources, like physical models or expert surveys, but often they are generated with statistical models. The second contribution of this thesis is the extension of isotonic distributional regression to allow covariates that are point predictions from a regression model, which may be trained on the same data to which isotonic distributional regression is to be applied. This combination yields a so-called distributional index model. Asymptotic consistency is proved under suitable assumptions, and real data applications demonstrate the usefulness of the method. Isotonic distributional regression provides a benchmark in forecasting problems, as it allows to quantify the merits of a specific, tailored model for the application at hand over a generic method which only relies on monotonicity. In such comparisons it is vital to assess the significance of forecast superiority or of forecast misspecification. The third contribution of this thesis is the development of new, safe methods for forecast evaluation, which require no or minimal assumptions on the data generating processes

Conditional quantile processes based on series or many regressors

Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from the data. We develop large sample theory for the empirical QR coefficient process, namely we obtain uniform strong approximations to the empirical QR coefficient process by conditionally pivotal and Gaussian processes, as well as by gradient and weighted bootstrap processes. We apply these results to obtain estimation and inference methods for linear functionals of the conditional quantile function, such as the conditional quantile function itself, its partial derivatives, average partial derivatives, and conditional average partial derivatives. Specifically, we obtain uniform rates of convergence, large sample distributions, and inference methods based on strong pivotal and Gaussian approximations and on gradient and weighted bootstraps. All of the above results are for function-valued parameters, holding uniformly in both the quantile index and in the covariate value, and covering the pointwise case as a by-product. If the function of interest is monotone, we show how to use monotonization procedures to improve estimation and inference. We demonstrate the practical utility of these results with an empirical example, where we estimate the price elasticity function of the individual demand for gasoline, as indexed by the individual unobserved propensity for gasoline consumption.

Statistical Contributions to Order Restricted Inference for Survival Data Analysis

This dissertation aims to study order restricted inference for survival data analysis where a hazard function is assumed to have a shape restriction with respect to continuous covariates. In the first chapter, we consider estimation of the semiparametric proportional hazards model with a completely unspecified baseline hazard function where the effect of a continuous covariate is assumed isotonic (or monotone) but otherwise unspecified. The pseudo iterative convex minorant algorithm is proposed to compute the isotonic estimator by optimizing a sequence of pseudo partial likelihood functions. A local consistency is established for a one-step update of the estimator when an initial value is in a shrinking neighborhood of the true value. Analysis of data from a recent HIV prevention study illustrates the practical utility of the methodology in estimating monotonic covariate effects that are nonlinear. In the second chapter, we consider additive hazards model with a unimodal hazard function in a continuous covariate with unknown mode. A quadratic loss function is defined, which allows efficient computations to estimate the mode and unimodal covariate effects. The methodology is applied to analyze the data from a recent randomized clinical trial of cardiovascular disease in kidney transplant patients. In the third chapter, we focus on multiple continuous covariates for a shape restricted hazard function. By assuming an additive isotonic structure of the multiple covariates under the proportional hazards model, the hazard function is defined as isotonic with respect to the partial order on the covariates. An efficient computation is proposed by combining the pseudo iterative convex minorant algorithm and the cycling algorithm. We use the proposed method to analyze the data from a recent clinical trial with cardiovascular outcome.Doctor of Philosoph

SEMIPARAMETRIC METHODS TO IMPROVE RISK ASSESSMENT AND DYNAMIC PREDICTION

Incorporating promising biomarkers to improve risk assessment and prediction is the central goal in many biomedical studies. Cost-effective designs and longitudinal designs are often utilized for measuring biomarker information, but they pose challenges to the data analyses. Statistical analyses for these kinds of data are routinely performed using parametric models. When the model assumptions are violated, parametric models may lead to substantial bias in parameter estimation, risk evaluation and prediction. In this dissertation, we will develop robust, exible statistical methods for risk assessment for matched case-control, nested case-control, and case-cohort designs, as well as a dynamic prediction tool for longitudinal data. In the first aim, we will develop a distribution-free method for identifying an optimal combination of biomarkers to differentiate cases and controls in matched case-control data. In the second aim, we will develop a semiparametric regression model with minimal assumptions on the link function for data from two-phase sampling designs with binary outcomes. In the third aim, we will develop a model-free dynamic prediction method for a survival outcome that provides dynamically updated risk scores using longitudinal biomarker(s)

Conditional Quantile Processes Based on Series or Many Regressors

Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from the data. We develop large sample theory for the empirical QR coefficient process, namely we obtain uniform strong approximations to the empirical QR coefficient process by conditionally pivotal and Gaussian processes, as well as by gradient and weighted bootstrap processes. We apply these results to obtain estimation and inference methods for linear functionals of the conditional quantile function, such as the conditional quantile function itself, its partial derivatives, average partial derivatives, and conditional average partial derivatives. Specifically, we obtain uniform rates of convergence, large sample distributions, and inference methods based on strong pivotal and Gaussian approximations and on gradient and weighted bootstraps. All of the above results are for function-valued parameters, holding uniformly in both the quantile index and in the covariate value, and covering the pointwise case as a by-product. If the function of interest is monotone, we show how to use monotonization procedures to improve estimation and inference. We demonstrate the practical utility of these results with an empirical example, where we estimate the price elasticity function of the individual demand for gasoline, as indexed by the individual unobserved propensity for gasoline consumption