Date of Award
Doctor of Philosophy (PhD)
Park , Chanseok
Sun , Xiaoqian
Taylor , Robert
This dissertation aims to address two problems in regression
analysis. One problem is the model selection and robust parameter estimation in high dimensional linear regressions. The other is concerning developing a robust and efficient estimator in nonparametric regressions.
In Chapter 1, we introduce the robust and efficient regression analysis, discuss those two interesting problems and our motivations, and present several exciting results.
We propose a novel robust penalized method for high dimensional linear regression in Chapter 2. Asymptotic properties are established and a data-driven procedure is developed to select adaptive penalties. We show it is the very first estimator to achieve desired oracle properties with certainty for high dimensional linear regression. Extensive simulations have been conducted and demonstrate the usefulness of the new technique.
A new local polynomial nonparametric regression is developed in Chapter 3. It minimizes a convex combination of several weighted loss functions simultaneously. The optimal weights are selected by a proposed procedure and adapt to the tails of the error distribution resulting in a procedure which is both robust and resistant. The asymptotic properties have been investigated. We show the resulting estimators are at least as efficient as those provided by existing procedures, but can be much more efficient for many distributions.
Its excellent finite sample performance is presented through simulations under a
variety of settings. A real data analysis exhibits the usefulness of the
Zheng, Qi, "Robust and Efficient Regression" (2013). All Dissertations. 1107.