A sharp adaptive confidence ball for self-similar functions

TL;DR

This paper introduces an adaptive confidence ball in a Gaussian sequence model that adjusts to unknown function smoothness and self-similarity, providing honest coverage with minimal assumptions.

Contribution

It constructs the first confidence ball that adapts to unknown smoothness and self-similarity, achieving optimal coverage in nonparametric Gaussian models.

Findings

Confidence ball adapts to unknown smoothness

Exact asymptotic coverage over self-similar spaces

Self-similarity condition is proven to be minimal

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.