We show that there do not exist adaptive confidence bands for curve estimation except under very restrictive assumptions. We propose instead to construct adaptive bands that cover a surrogate function f⋆ which is close to, but simpler than, f. The surrogate captures the significant features in f. We establish lower bounds on the width for any confidence band for f⋆ and construct a procedure that comes within a small constant factor of attaining the lower bound for finite-samples.
"Adaptive confidence bands." Ann. Statist. 36 (2) 875 - 905, April 2008. https://doi.org/10.1214/07-AOS500