Empirical Bernstein Confidence Intervals for Kernel Smoothers: A Safe and Sharp Way to Exhaust Assumed Smoothness

TL;DR

This paper introduces empirical Bernstein confidence intervals for kernel smoothers that effectively control bias and stochastic fluctuations, providing accurate coverage and optimal width without normalization issues.

Contribution

It develops a new calibration method using empirical Bernstein tail control that improves inference for kernel smoothers by avoiding bias amplification from normalization.

Findings

Intervals attain nominal coverage up to a small remainder

Widths shrink at the minimax rate $n^{-S/(2S+1)}$

Method applies to univariate density and regression functions

Abstract

Using normal approximation (NA) to construct confidence intervals for kernel smoothers faces a fundamental challenge: the normalization that produces a limiting distribution also magnifies smoothing bias, so that a small estimation bias may become a non-negligible inferential bias. Robust bias correction (RBC) and bias-aware inference (BA) address this difficulty through different bias-control strategies. This paper takes a different route by replacing the normal-approximation calibration engine with empirical Bernstein tail control. The resulting confidence intervals control stochastic fluctuations on the original estimation scale, so that deterministic smoothing bias enters the radius as an estimation-scale approximation error rather than as a normalized inferential bias. We develop this idea for pointwise inference on univariate density and regression functions. The proposed empirical Bernstein confidence intervals (EBCIs) combine empirical Bernstein calibration with bias-aware fixed-length radius construction under a local Taylor-remainder class. Uniformly over functions with \(S\)-th order local smoothness, both one-sided and two-sided intervals attain the nominal coverage level up to a remainder of order $n^{-\frac{2S}{2S+1}}$, or an exponential remainder in bounded or sub-Gaussian settings. Their widths shrink at the minimax rate $n^{-\frac{S}{2S+1}}$. Thus, EBCI safely converts correctly specified smoothness into both coverage accuracy and interval-length efficiency. The contribution is not a new bias-control philosophy, but a new calibration engine that can inherit existing ideas such as BA and RBC while avoiding the usual normalization-induced amplification of smoothing bias.