  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.
