Optimal bandwidths for local polynomial regression usually involve functionals of the derivatives of the unknown regression function. In the multivariate case, estimates of these functionals are not readily available, primarily because estimating multivariate derivatives is complicated. In this paper, an estimator of multivariate second derivative is obtained via local quadratic regression with cross terms left out. This estimator has the optimal rate of convergence but is simpler and uses a lot less computing time than the full local quadratic estimator. Using this as a pilot estimator, an estimator of the integrated squared Laplacian of a multivariate regression function is obtained which leads to a plug-in formula of the optimal bandwidth for multivariate local linear regression. This bandwidth has good theoretical properties as well as satisfactory performance in our simulation study. It is also recommended for variable selection methods.
Bandwidth selection Asymptotic optimality Functional estimation Laplacian Local quadratic regression Second derivatives Within bias trade off