A sharp adaptive confidence ball for self-similar functions

Abstract

In the nonparametric Gaussian sequence space model an ${\ell }^2$-confidence ball $C_n$ is constructed that adapts to unknown smoothness and Sobolev-norm of the infinite-dimensional parameter to be estimated. The confidence ball has exact and honest asymptotic coverage over appropriately defined `self-similar' parameter spaces. It is shown by information-theoretic methods that this `self-similarity' condition is weakest possible.

To appear in Stochastic Processes and Applications (memorial issue for E. Giné)