Multiparameter Bandwidth Processes and Adaptive Surface Smoothing

We derive a functional limit theorem for a sequence of bandwidth processes with multivariate time and show that the limit process is multivariate Gaussian. This theorem is then applied to show asymptotic efficiency of certain data-adaptive local bandwidth choices for kernel estimators of multivariate regression functions and their derivatives. The cases where optimal multivariate bandwidths exist as minimizers of leading mean squared error terms are characterized.