Adaptation bounds for confidence bands under self-similarity

We derive bounds on the scope for a confidence band to adapt unknown regularity of nonparametric function that is observed with noise, such as regression or density, under self-similarity condition proposed by Giné and Nickl (Ann. Statist. 38 (2010) 1122–1170). find adaptation can only be achieved up term depends choice constant used define self-similarity, this becomes arbitrarily large conservative choices constant. construct achieves bound, does not depend Our results suggest care must taken in choosing interpreting defines since dependence adaptive bands cannot made disappear asymptotically.