Thesis (Ph. D.)--State University of New York at Binghamton, Department of Mathematical Sciences, 2005. / Includes bibliographical references.

Thesis (M.S.)--Baylor University, 2006. / Includes bibliographical references (p. 77-80).

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (leaves 147-153).

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 140-145).

This thesis explores approaches to regression that utilise the treatment of covariates as random variables. The distribution of covariates, along with the conditional regression model Y | X, define the joint model over (Y,X), and in particular, the marginal distribution of the response Y. This marginal distribution provides a vehicle for the incorporation of prior information, as well as external, marginal data. The marginal distribution of the response provides a means of parameterisation that can yield scalable inference, simple prior elicitation, and, in the case of survival analysis, the complete treatment of truncated data. In many cases, this information can be utilised without need to specify a model for X. Chapter 2 considers the application of Bayesian linear regression where large marginal datasets are available, but the collection of response and covariate data together is limited to a small dataset. These marginal datasets can be used to estimate the marginal means and variances of Y and X, which impose two constraints on the parameters of the linear regression model. We define a joint prior over covariate effects and the conditional variance σ² via a parameter transformation, which allows us to guarantee these marginal constraints are met. This provides a computationally efficient means of incorporating marginal information, useful when incorporation via the imputation of missing values may be implausible. The resulting prior and posterior have rich dependence structures that have a natural 'analysis of variance' interpretation, due to the constraint on the total marginal variance of Y. The concept of 'marginal coherence' is introduced, whereby competing models place the same prior on the marginal mean and variance of the response. Our marginally constrained prior can be extended by placing priors on the marginal variances, in order to perform variable selection in a marginally coherent fashion. Chapter 3 constructs a Bayesian nonparametric regression model parameterised in terms of FY , the marginal distribution of the response. This naturally allows the incorporation of marginal data, and provides a natural means of specifying a prior distribution for a regression model. The construction is such that the distribution of the ordering of the response, given covariates, takes the form of the Plackett-Luce model for ranks. This facilitates a natural composite likelihood approximation that decomposes the likelihood into a term for the marginal response data, and a term for the probability of the observed ranking. This can be viewed as a extension to the partial likelihood for proportional hazards models. This convenient form leads to simple approximate posterior inference, which circumvents the need to perform MCMC, allowing scalability to large datasets. We apply the model to a US Census dataset with over 1,300,000 data points and more than 100 covariates, where the nonparametric prior is able to capture the highly non-standard distribution of incomes. Chapter 4 explores the analysis of randomised clinical trial (RCT) data for subgroup analysis, where interest lies in the optimal allocation of treatment D(X), based on covariates. Standard analyses build a conditional model Y | X,T for the response, given treatment and covariates, which can be used to deduce the optimal treatment rule. We show that the treatment of covariates as random facilitates direct testing of a treatment rule, without the need to specify a conditional model. This provides a robust, efficient, and easy-to-use methodology for testing treatment rules. This nonparametric testing approach is used as a splitting criteria in a random-forest methodology for the exploratory analysis of subgroups. The model introduced in Chapter 3 is applied in the context of subgroup analysis, providing a Bayesian nonparametric analogue to this approach: where inference is based only on the order of the data, circumventing the requirement to specify a full data-generating model. Both approaches to subgroup analysis are applied to data from an AIDS Clinical Trial.

With the development of computing and sampling technologies, high dimensionality has become an important characteristic of commonly used science data, such as some data from bioinformatics, information engineering, and the social sciences. The varying coefficient model is a flexible and powerful statistical model for exploring dynamic patterns in many scientific areas. It is a natural extension of classical parametric models with good interpretability, and is becoming increasingly popular in data analysis. The main objective of thesis is to apply the varying coefficient model to analyze high dimensional data, and to investigate the properties of regularization methods for high-dimensional varying coefficient models. We first discuss how to apply local polynomial smoothing and the smoothly clipped absolute deviation (SCAD) penalized methods to estimate varying coefficient models when the dimension of the model is diverging with the sample size. Based on the nonconcave penalized method and local polynomial smoothing, we suggest a regularization method to select significant variables from the model and estimate the corresponding coefficient functions simultaneously. Importantly, our proposed method can also identify constant coefficients at same time. We investigate the asymptotic properties of our proposed method and show that it has the so called “oracle property.” We apply the nonparametric independence Screening (NIS) method to varying coefficient models with ultra-high-dimensional data. Based on the marginal varying coefficient model estimation, we establish the sure independent screening property under some regular conditions for our proposed sure screening method. Combined with our proposed regularization method, we can systematically deal with high-dimensional or ultra-high-dimensional data using varying coefficient models. The nonconcave penalized method is a very effective variable selection method. However, maximizing such a penalized likelihood function is computationally challenging, because the objective functions are nondifferentiable and nonconcave. The local linear approximation (LLA) and local quadratic approximation (LQA) are two popular algorithms for dealing with such optimal problems. In this thesis, we revisit these two algorithms. We investigate the convergence rate of LLA and show that the rate is linear. We also study the statistical properties of the one-step estimate based on LLA under a generalized statistical model with a diverging number of dimensions. We suggest a modified version of LQA to overcome its drawback under high dimensional models. Our proposed method avoids having to calculate the inverse of the Hessian matrix in the modified Newton Raphson algorithm based on LQA. Our proposed methods are investigated by numerical studies and in a real case study in Chapter 5.

No description available.

In this dissertation, we investigate various theoretical properties of optimal regression designs and develop several numerical algorithms for computing them. The results can be applied to linear, nonlinear and generalized linear models. Our work starts from how to solve the design problems for A-, As, c-, I- and L-optimality criteria on one-response model. Theoretical results are hard to derive for many regression models and criteria, and existing numerical algorithms can not compute the results efficiently when the number of support points is large. Therefore we consider to solve the design problems based on SeDuMi program in MATLAB. SeDuMi is developed to solve semidefinite programming (SDP) problems in optimization. To apply it, we derive a general transformation to connect the design problems with SDP problems, and propose a numerical algorithm based on SeDuMi to solve these SDP problems. The algorithm is quite general under the least squares estimator (LSE) and weighted least squares estimator (WLSE) and can be applied to both linear and nonlinear regression models. We continue to study the optimal designs based on one-response model when the error distribution is asymmetric. Since the second-order least squares estimator (SLSE) is more efficient than the LSE when the error distribution is not symmetric, we study optimal designs under the SLSE. We derive expressions to characterize A- and D-optimality criteria and develop a numerical algorithm for finding optimal designs under the SLSE based on SeDuMi and CVX programs in MATLAB. Several theoretical properties are also derived for optimal designs under SLSE. To check the optimality of the numerical results, we establish the Kiefer-Wolfowitz equivalence theorem and apply it to various applications. Finally, we discuss the optimal design problems for multi-response models. Our algorithm studied here is based on SeDuMi and CVX, and it can be used for linear, nonlinear and generalized linear models. The transformation invariance property and dependence on the covariance matrix of the correlated errors are derived. We also correct the errors in the literature caused by formulation issues. The results are very useful to construct optimal regression designs on discrete design space. They can be applied to any one-response and multi-response models, various optimality criteria, and several estimators including LSE, maximum likelihood estimator, best linear unbiased estimator, SLSE and WLSE. / Graduate

Please read the abstract in the dissertation. / Dissertation (MSc)--University of Pretoria, 2011. / Statistics / unrestricted

Regression models with a scalar response and a functional predictor have been extensively studied. One approach is to approximate the functional predictor using basis function or eigenfunction expansions. In the expansion, the coefficient vector can either be fixed or random. The random coefficient vector is also known as random effects and thus the regression models are in a mixed effects framework. The random effects provide a model for the within individual covariance of the observations. But it also introduces an additional parameter into the model, the covariance matrix of the random effects. This additional parameter complicates the covariance matrix of the observations. Possibly, the covariance parameters of the model are not identifiable. We study identifiability in normal linear mixed effects models. We derive necessary and sufficient conditions of identifiability, particularly, conditions of identifiability for the regression models with a scalar response and a functional predictor using random effects. We study the regression model using the eigenfunction expansion approach with random effects. We assume the random effects have a general covariance matrix and the observed values of the predictor are contaminated with measurement error. We propose methods of inference for the regression model's functional coefficient. As an application of the model, we analyze a biological data set to investigate the dependence of a mouse's wheel running distance on its body mass trajectory. / Science, Faculty of / Statistics, Department of / Graduate