Jeffrey Pitblado, Southern Methodist University
Evaluation: Estimating Partially Variable Bandwidths in Local
Linear Regression using an Information CriterionThere 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
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