Safe and Sharp Honest Inference for Nonparametric Estimation via Empirical Bernstein Calibration

Abstract

Constructing honest confidence intervals often depends on reliable bias control and accurate calibration based on asymptotic normality, including standard-normal and folded-normal calibration. Substantial progress has been made in correcting or controlling smoothing bias, including robust bias correction and bias-aware inference. We first show that, even after smoothing bias has been well corrected or controlled, asymptotic-normality-based calibration may still be a binding source of finite-sample undercoverage. Thus, the resulting intervals may struggle to achieve the minimax shrinkage rate and uniformly small undercoverage error simultaneously. Instead of using distributional approximation, we calibrate the radius directly by combining an empirical Bernstein bound, a data-driven variance proxy, Lepski-type bandwidth selection, and a bias-aware fixed-length-radius criterion. The formal theory covers nonparametric regression and density estimation, with regression results ranging from local-polynomial to sieve estimators. The resulting empirical Bernstein confidence intervals are safe and sharp. Uniformly over functions with $S$-th order local smoothness, both one-sided and two-sided intervals attain nominal coverage up to $o(n^{-2S/(2S+1)})$, or exponential remainders under bounded or sub-Gaussian conditions, while their widths shrink at the minimax rate $n^{-S/(2S+1)}$ (or up to a $\sqrt{\log n}$-level factor). The calibration principle is modular and can also be combined with other existing bias-control strategies, like robust bias correction. Thus, the contribution of this paper is not a bias-control device but a new angle of calibration. Compared with asymptotic-normality-based calibration, empirical Bernstein calibration safely and conveniently converts the specified smoothness into both coverage accuracy and interval-length efficiency. Simulations support the theory.