Jeffrey Pitblado, Southern Methodist University

Evaluation: Estimating Partially Variable Bandwidths in Local Linear Regression using an Information Criterion

There are many problems that require the estimation of the expected value of a response variable given a predictor variable, while making as few assumptions on the shape of the regression function as possible. Many approaches to such problems use some form of linear smoothing. The more popular ones employ kernel weight functions or weighted least squares local polynomial fits. All of these smoothing techniques require a smoothing parameter, also known as a bandwidth, which essentially controls the amount of bias and variance in the mean function estimate. Bandwidth selection is a hard problem. It is particularly difficult to choose a bandwidth when the underlying curvature of the mean function is not reasonably constant. Global bandwidths are known to have poor performance in this situation, while fully variable bandwidths require as many smoothing parameters as data points. An alternative type of bandwidth holds some promise in this case; a partially variable bandwidth (PVBW) is a piece-wise constant function defined on a partition of the range of observed predictors. This provides more flexibility than a global bandwidth and contains fewer parameters than a fully variable bandwidth. This dissertation develops a data-based methodology for choosing a PVBW, in the local linear regression setting. This technique uses an improved Akaike Information Criterion originally introduced for model selection in multiple regression and time series. Optimization at two levels is investigated. The first level selects the constants in a PVBW for a fixed partition number. The second level involves choosing a parsimonious partition number, which is particularly challenging. Some adjustments to the methodology are proposed and evaluated using average squared error calculations in a Monte Carlo simulation study.