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

Using normal approximation (NA) to construct a kernel-smoother-based confidence interval faces a fundamental challenge: the normalization makes a small estimation bias become a non-negligible inferential bias. This paper takes a different route by replacing the normal-approximation-based tail control with empirical Bernstein tail control. The resulting confidence intervals control stochastic variability 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 $n^{-\frac{S}{2S+1}}$. Moreover, in the small-$\alpha$ regime, the EBCI radius is first-order aligned with the radii of bias-aware-type fixed-length confidence intervals. For one-sided inference, the leading term coincides, while for two-sided inference the only difference is the usual replacement of $\log(\frac{1}{\alpha})$ by $\log(\frac{2}{\alpha})$. Thus, EBCI safely converts correctly specified smoothness into both coverage accuracy and interval-length efficiency. The contribution is not a new bias-control approach, but a new calibration method that can inherit existing ideas such as bias-aware inference (BA) and robust bias correction (RBC) while avoiding the usual normalization-induced amplification of smoothing bias.