Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations

TL;DR

This paper introduces novel multivariate matching estimators that achieve root-n consistency without smoothing parameters, addressing bias issues in high-dimensional settings for inverse density weighted expectations.

Contribution

It develops polynomial least squares-based matching estimators on Voronoi cells that converge at the parametric rate without requiring density smoothness or smoothing parameters.

Findings

Estimators achieve root-n convergence rate in multivariate settings.

Proposed methods do not rely on nonparametric density estimation.

Simulations confirm practical effectiveness.

Abstract

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.