Title: Some Topics in Nonparametric Regression

Abstract:

This dissertation studies variance estimation, smoothing-parameter selection, and locally adaptive smoothing in multidimensional nonparametric regression. It begins by developing residual-variance estimators for two-dimensional fixed grids and random designs, with theoretical and empirical evaluation. The variance estimators exhibit function-dependent bias–variance tradeoffs; adaptive local polynomial estimators are especially effective for nonlinear functions in the random-design simulations. The dissertation then derives an exact block-deletion identity for Gaussian radial basis function smoothing and compares repeated five-fold cross-validation with repeated 80–20 holdout. Repeated five-fold cross-validation generally selects the regularization coefficient more stably on a fixed dataset, although neither has a clear unconditional regression-function estimation advantage. Finally, it extends a combination-of-estimates approach by selecting Gaussian RBF candidates pointwise using estimated local risk. The adaptive estimator improves on global GCV in all 12 two-dimensional and 12 of 15 five-dimensional configurations, particularly for localized peaks and corner behavior. Real-data applications illustrate the fixed- and random-design implementations.