A locally weighted least squares approach to nonparametric regression

Papers on kernel estimators have appeared in the statistical literature for nearly two decades. Much of the research has focused on finding kernels which optimize the performance of the estimator. However, given that the kernel estimator can be viewed as a locally weighted least squares estimator using a zero degree polynomial, it is surprising how little has been devoted to extending research to locally weighted least squares estimators with higher degree polynomials.;This dissertation explores locally weighted least squares as a nonparametric regression estimator. Results are presented for estimation using either one or two predictor variables, and for arbitrary degree of local polynomial. In particular, asymptotic mean squared error rates of convergence are presented which apply to estimation in the interior and the boundary regions. These results indicate that locally weighted least squares estimation performs better in the boundary region than typical kernel estimators because they do not suffer from boundary bias as severely. In fact, increasing the degree of the local polynomial used for regression yields higher rates of convergence throughout the interval of estimation.;A method of selecting a local bandwidth parameter from the data is also studied. Implementation of such a bandwidth selection technique allows complete automation of the estimation procedure once the degree of the local polynomial has been decided. This bandwidth selection method is adapted from a technique developed for kernel estimators by Eubank and Schucany (1992). It attempts to balance the dominant bias squared and variance terms given in the asymptotic mean squared error expression for the estimator.;Simulations are performed to evaluate the finite sample performance of the locally weighted least squares estimator under a variety of conditions. Comparison is made to simulations for a kernel estimator using the same data and the results are discussed. Finally, it is shown that locally weighted least squares estimators are asymptotically equivalent to kernel estimators for appropriately chosen kernel functions. Similar relationships also hold for locally weighted least squares estimators having local polynomials of different degrees. These relationships are investigated and a method for obtaining asymptotically equivalent kernels and weight functions is presented.