Approximations to the mean integrated squared error with applications to optimal bandwidth selection for nonparametric regression function estimators

Discrete versions of the mean integrated squared error (MISE) provide stochastic measures of accuracy to compare different estimators of regression fuctions. These measures of accuracy have been used in Monte Carlo trials and have been employed for the optimal bandwidth selection for kernel regression function estimators, as shown in [5], Optimal Bandwidth Selection in Nonparametric Regression Function Estimation. Inst. of Statistics Mimeo Series No. 1530, Univ. of North Carolina, Chapel Hill). In the present paper it is shown that these stochastic measures of accuracy converge to a weighted version of the MISE of kernel regression function estimators, extending a result of [6], Biometrika 69, 383-390) and Marron (1983, J. Multivariate Anal. 18, No. 2) to regression function estimation.