Contributions to level set estimation, nonparametric regression, confidence intervals from imputed data and marginal logistic regression models for longitudinal survey dat

The objective of the thesis is to develop and extend some new results on four selected topics which are currently widely used and studied in probability and statistics. These topics include: (1) Nonparametric estimators of level sets and their properties under minimal conditions. We establish the same results as in Bailo et al. (2001) without imposing their technical condition. Furthermore, the IID assumption will be relaxed to one based on alpha-mixing sequence. We require no additional conditions to establish our results. (2) The convergence rates for nonparametric regression. Motivated by the classical results on this topic for the IID case, I extended some of the results to dependent mixing sequences. For this topic, I studied two problems, one is about the rate of convergence for complexity regularization with beta-mixing data; the other one is the convergence rate in nonparametric regression for an alpha-mixing setup. Our technical arguments are based on the theory of function-indexed empirical processes and a strong coupling lemmas. (3) Confidence intervals with fractional imputed missing data. For missing data set, fractional imputed data are used and confidence intervals are constructed for population parameters such as mean, distribution and quantile via normal approximation and empirical likelihood methods. To do so, some asymptotic normality theorems need to be set up. These results are extension of Qin, Rao and Ren's (2006) work for random hot deck imputation. We also consider two cases for this work: one is based on simple data sets without any covariate; the other is assuming that observation and some covariates are available and there is a linear relationship between them. (4) Marginal logistic regression for longitudinal complex survey design data. The motivation of the study comes from the requirement of processing of Statistics Canada conducted longitudinal survey data---the National Population Healthy Survey (NPHS). We proposed a Generalized Estimation Equation (GEE) method for the estimation of model parameters. To avoid the complex derivation of Taylor linearization, one-step estimation function (EF) bootstrap method is used for the variance estimation. Several Goodness-of-fit tests are studied for the model assessment problems.;To verify and support the theoretical derivation in the work, a large amount of computational simulation work has been done with C/C++ and SAS code. Some of the simulation results are listed in the thesis.