| We consider the kernel-type estimators for obtaining the nonparametric estimate of a density function and a regression function. Specifically, in this dissertation we study the properties of the kernel estimator of the unknown curve (density or regression function) with the local bandwidth selections. Under weak conditions, we prove that the local kernel estimators uniformly converge to the unknown curve with probability one. Also, we calculate the convergence rate of this kind of estimators. For optimal bandwidth selections, the local bandwidth estimators are superior to the global bandwidth estimators with respect to the integrated mean squared error of the estimators. We present a simulation as a test of the method with the local bandwidth estimators. |
