Model selection and nonparametric estimation for regression models - He, Zonglin, author; Opsomer, Jean, advisor; Breidt, F. Jay, committee member; Meyer, Mary, committee member; Elder, John, committee member (Colorado State University. Libraries, 2014)
In this dissertation, we deal with two different topics in statistics. The first topic in survey sampling deals with variable selection for linear regression model from which we will sample with a possibly informative design. Under the assumption that the finite population is generated by a multivariate linear regression model from which we will sample with a possibly informative design, we particularly study the variable selection criterion named predicted residual sums of squares in the sampling context theoretically. We examine the asymptotic properties of weighted and unweighted predicted residual sums of squares under weighted least squares regression estimation and ordinary least squares regression estimation. One simulation study for the variable selection criteria are provided, with the purpose of showing their ability to select the correct model in the practical situation. For the second topic, we are interested in fitting a nonparametric regression model to data for the situation in which some of the covariates are categorical. In the univariate case where the covariate is a ordinal variable, we extend the local polynomial estimator, which normally requires continuous covariates, to a local polynomial estimator that allows for ordered categorical covariates. We derive the asymptotic conditional bias and variance for the local polynomial estimator with ordinal covariate, under the assumption that the categories correspond to quantiles of an unobserved continuous latent variable. We conduct a simulation study with two patterns of ordinal data to evaluate our estimator. In the multivariate case where the covariates contain a mixture of continuous, ordinal, and nominal variables, we use a Nadaraya-Watson estimator with generalized product kernel. We derive the asymptotic conditional bias and variance for the Nadaraya-Watson estimator with continuous, ordinal, and nominal covariates, under the assumption that the categories of the ordinal covariate correspond to quantiles of an unobserved continuous latent variable. We conduct a multivariate simulation study to evaluate our Nadaraya-Watson estimator with generalized product kernel.