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

Abstract

Calibration of an honest confidence interval means choosing, for each $α\in(0,1)$, how the corresponding $α$-critical value is converted into a radius yielding coverage probability at least $1-α$. Standard-normal critical-value calibration (SNC) is the default route for many confidence intervals based on nonparametric smoothers in nonparametric econometrics. However, this calibration method creates a structural difficulty: the normalization yielding a limiting distribution also makes a small estimation bias become a non-negligible inferential bias. We take a different calibration route by combining the tail control of empirical Bernstein inequalities with a fixed-length-radius optimization from bias-aware inference. We establish the formal theory in canonical scalar-covariate regression and density settings, with the regression theory ranging from local-polynomial to weighted-average estimators. The resulting empirical Bernstein confidence intervals (EBCIs) are "safe" and "sharp". Safety means that, uniformly over functions with some $S$-th order local smoothness, both one-sided and two-sided intervals attain the nominal coverage level up to a remainder $o(n^{-\frac{2S}{2S+1}})$, or an exponential remainder in bounded or sub-Gaussian settings. Sharpness means that interval widths shrink at the minimax rate $n^{-\frac{S}{2S+1}}$. Moreover, in the small-$α$ regime, the EBCI radius is first-order aligned with the radii of bias-aware fixed-length confidence intervals. 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 principle for the radius of a confidence interval. The method can be combined with existing ideas such as bias-aware inference (BA) and robust bias correction (RBC), while avoiding the bias inflation induced by SNC.