  Expected values weighted by the inverse of a multivariate density or,
equivalently, Lebesgue integrals of regression functions with multivariate
regressors occur in various areas of applications, including estimating average
treatment effects, nonparametric estimators in random coefficient regression
models or deconvolution estimators in Berkson errors-in-variables models. The
frequently used nearest-neighbor and matching estimators suffer from bias
problems in multiple dimensions. By using polynomial least squares fits on each
cell of the $K^{\text{th}}$-order Voronoi tessellation for sufficiently large
$K$, we develop novel modifications of nearest-neighbor and matching estimators
which again converge at the parametric $\sqrt n $-rate under mild smoothness
assumptions on the unknown regression function and without any smoothness
conditions on the unknown density of the covariates. We stress that in contrast
to competing methods for correcting for the bias of matching estimators, our
estimators do not involve nonparametric function estimators and in particular
do not rely on sample-size dependent smoothing parameters. We complement the
upper bounds with appropriate lower bounds derived from information-theoretic
arguments, which show that some smoothness of the regression function is indeed
required to achieve the parametric rate. Simulations illustrate the practical
feasibility of the proposed methods.
