Optimal bandwidth choice for density-weighted averages

This paper characterizes the optimal bandwidth value for estimating density-weighted averages, statistics that arise in semiparametric estimation methods for index models and models of selected samples based on nonparametric kernel estimators. The optimal bandwidth is derived by minimizing the leading terms of mean squared error of the density-weighted average. The optimal bandwidth formulation is developed by comparison to the optimal pointwise bandwidth of a naturally associated nonparametric estimation problem, highlighting the role of sample size and the structure of nonparametric estimation bias. The methods are illustrated by estimators of average density, density-weighted average derivatives, and conditional covariances, and bandwidth values are calculated for normal designs. A simple 'plug-in' estimator for the optimal bandwidth is proposed. Finally, the optimal bandwidth for estimating ratios of density-weighted averages is derived, showing that the earlier optimal formulae can be implemented directly using naturally defined 'residual' values.