A central goal of causal inference is to detect and estimate the treatment effects of a given treatment or intervention on an outcome variable of interest, where a member known as the heterogeneous treatment effect (HTE) is of growing popularity in recent practical applications such as the personalized medicine. In this paper, we model the HTE as a smooth nonparametric difference between two less smooth baseline functions, and determine the tight statistical limits of the nonparametric HTE estimation as a function of the covariate geometry. In particular, a two-stage nearest-neighbor-based estimator throwing away observations with poor matching quality is near minimax optimal. We also establish the tight dependence on the density ratio without the usual assumption that the covariate densities are bounded away from zero, where a key step is to employ a novel maximal inequality which could be of independent interest.

Summary and Contributions: The purpose of this paper is to provide new theoretical tools and bounds for the heterogeneous treatment effect (HTE) estimation in causal inference. This work is in line with a fairly current theme: the HTE estimation is experiencing a growing interest in applications, particularly in the field of personalized medicine. To avoid strong assumptions and to benefit from a broader scope of application, the authors focus on nonparametric estimation. As the authors point out, much effort has been devoted to proposing practical methods, but not so much to the statistical study of nonparametric HTE estimation. This paper establishes minimax rates with dependence on both the geometry of the covariates, and parameters related to propensity scores and noise levels.

The knowledgeable reviewers agree that this is a good paper that warrants acceptance. There was no major concern raised during the discussion phase and the rebuttal has further supported the vote for acceptance. The paper is therefore accepted as a spotlight. If the authors have time and agree that it will be beneficial, the important improvement for the paper is to add a simple experiment that demonstrates the effectiveness of the proposed estimator.

A central goal of causal inference is to detect and estimate the treatment effects of a given treatment or intervention on an outcome variable of interest, where a member known as the heterogeneous treatment effect (HTE) is of growing popularity in recent practical applications such as the personalized medicine. In this paper, we model the HTE as a smooth nonparametric difference between two less smooth baseline functions, and determine the tight statistical limits of the nonparametric HTE estimation as a function of the covariate geometry. In particular, a two-stage nearest-neighbor-based estimator throwing away observations with poor matching quality is near minimax optimal. We also establish the tight dependence on the density ratio without the usual assumption that the covariate densities are bounded away from zero, where a key step is to employ a novel maximal inequality which could be of independent interest.