Doctor of Philosophy
Braun, W. John
When performing local polynomial regression (LPR) with kernel smoothing, the choice of the smoothing parameter, or bandwidth, is critical. The performance of the method is often evaluated using the Mean Square Error (MSE). Bias and variance are two components of MSE. Kernel methods are known to exhibit varying degrees of bias. Boundary effects and data sparsity issues are two potential problems to watch for. There is a need for a tool to visually assess the potential bias when applying kernel smooths to a given scatterplot of data. In this dissertation, we propose pointwise confidence intervals for bias and demonstrate a software tool to implement the confidence bands in practice. The effectiveness of the proposed bias assessment tool is demonstrated using simulated data and is illustrated by its application to classical data sets. To reduce the bias of LPR while keeping other good properties of it, as well as to mitigate the sparsity and boundary issues, this thesis extends the technique of double-smoothing from the local linear regression context to higher order local polynomial regression, with particular focus on local quadratic and local cubic regression. Double-smoothing is a technique that involves two levels of smoothing and is known to reduce bias in nonparametric regression while maintaining control over variance. What was not known is that the method is often sub-optimal when the bandwidths at both levels of smoothing are equal. In this thesis, we propose a simple method for obtaining the second-level smoothing bandwidth while using a cross-validation method for the first level. The proposed tools in this dissertation are employed to address real problems related to wildfire management: one is for modelling the time to initial attack using fire case records for the years from 1930 to 2012 in a Northeastern Ontario area. Another one is for the experimental results of burning debris as a randomized component in a firebrand spotting simulator
Ma, Wenkai, "Bias Assessment and Reduction in Kernel Smoothing" (2018). Electronic Thesis and Dissertation Repository. 5901.