PhD Dissertation Abstracts: 2011PhD Dissertation Abstracts: 2011
Statistics PhD Alumni 2011:
Rongqi Chen (2011)
ADVISER: Wolfgang Polonik
TITLE: Asymptotic distribution for the plug-in estimation of level sets
ABSTRACT: In the area of nonparametric statistics, level set estimation received a lot of attention recently. Two main approaches are considered in literature: plug-in estimation and direct estimation. Our focus in this thesis is on the plug-in estimation in three different cases: density level set, regression level set and classification.
Mason and Polonik (2009) established a central limit theorem for the plugin estimator of the density level set under certain conditions. Inspired by some geometrical intuition, we will extend their result by using weaker assumptions. This result shows that there is some phase transition.
Our main contributions are on the regression level set estimation. The asymptotic behavior of the plug-in estimator of the regression level set will be studied thoroughly. Under suitable conditions, we will derive both the consistency and asymptotic normality, the rates of which are similar to the density case.
The case of binary classification can be considered as a special case of regression. Using the results from the regression case, we also develop the consistency and asymptotic normality for the misclassification error. However, since the expectation for the misclassification error of the plug-in classifier depends on unknown quantities, the asymptotic normality cannot be used for statistical inferences directly in practice. One approach to tackle this problem is based on the idea of data splitting, which is also studied in this thesis.
Lawrence Lee (2011)
ADVISER: Jiming Jiang
TITLE: Iterative Estimation Equation Approach for Nonlinear Mixed Effects Models
ABSTRACT: Nonlinear Mixed Effect Model (NLME) is a useful tool for analyzing pharmacokinetic data and other repeated measures data. Compared to Linear Mixed Effects Models (LME), Nonlinear mixed effects models have more flexilibility because it allows longitudinal response measurement to have a nonlinear relationship with the predictor vector. Maximum likelihood approach can be used to find the estimates of the parameters, which is usually shown to be consistent in linear models. In Nonlinear mixed effects models, because the random effects enter the model nonlinearly, in general, the marginal density of the response does not have a closed-form expression, and hence no closed-form exists for the likelihood. A natural solution to solve this problem is to implement approximation methods for the likelihood function. However, some of these approaches have a potential of producing inconsistent estimates when the variability in random effects is large.
An Iterative Estimation Equations (IEE) approach has been recently studied for a semiparametric regression model for the longitudinal data with unspecified covariance matrix; consistency and asymptotic efficiency have also been demonstrated. We extend this approach to Nonlinear mixed effect models. Simulations and case studies are conducted to illustrate the proposed estimation procedures.
Travis Loux (2011)
ADVISER: Christiana Drake
TITLE: Causal Inference and Estimation of the Odds Ratio
ABSTRACT: In this presentation we will explore two areas of causal inference: adjusting for confounding variables in observational studies and missing data in estimation of the odds ratio. We extend work developed under assumptions of linearity and additivity of treatment effects to binary models, where many properties of linear models do not hold. We will begin by reviewing the counterfactual model and looking at uses of propensity scores to adjust for confounding variables when trying to estimate a marginal odds ratio. We discuss stratification and weighting by the propensity score along with doubly robust estimators. We suggest the use of weighted estimators over stratified estimators for unbiased estimation of the marginal odds ratio. In the second part of the talk, we look at adjusting for non-ignorable missing data when data for a small subset of non-responders has been acquired through a less comprehensive follow-up survey. We compare multiple imputation methods with combining odds ratios from subgroups and show that combining odds ratios has the potential to decrease, though not completely eliminate, bias in estimating a collapsed odds ratio.
Michael McAssey (2011)
ADVISER: Fushing Hsieh
TITLE: Topics on associations among random processes
ABSTRACT: This dissertation focuses on innovative techniques for the statistical analysis of concurrent random phenomena. In particular, new methods are presented for estimating measures of the relationships among multiple random variables or processes based on data generated from a common population. Of special interest are multiple random processes, in which the relationships among data sources are in constant transition among several states or dierent levels of the same state. Such a scenario is considered in the analysis of EEG data recorded from the brain of a rat, in which a technique is developed to measure instants of coupling between pairs of signals generated from different brain regions, and model the evolution of brain activity in terms of several instantaneous coupling states. A similar idea is employed to measure the evolution of synchrony in physiological measures among couples while performing assigned tasks. An innovative method is designed for measurement of the linear association between two variables in the presence of heteroscedastic measurement error, and in the process a clever test for the misapplication of linear models is presented. The ben ts of the improved model are illustrated using public health data. This idea is also extended to the aforementioned study of synchrony in physiological measures. These methodologies provide new tools for scientists engaged in research well beyond the applications to neuroscience, public health and psychology highlighted in this dissertation. Finally, a study involving a specified mode of signal transmission iv in networks is presented.