Adaptive confidence sets in L^2

TL;DR

This paper develops methods for constructing adaptive confidence sets in L^2 for Sobolev class densities, identifying when full adaptation is possible and providing new theoretical bounds.

Contribution

It introduces a general nonparametric minimax testing approach and establishes new lower bounds for adaptive confidence sets in L^2.

Findings

Full adaptation is possible in certain parameter regimes.

Critical regions must be removed for adaptation in some cases.

New lower bounds for L^2-adaptive confidence sets are derived.

Abstract

The problem of constructing confidence sets that are adaptive in L^2-loss over a continuous scale of Sobolev classes of probability densities is considered. Adaptation holds, where possible, with respect to both the radius of the Sobolev ball and its smoothness degree, and over maximal parameter spaces for which adaptation is possible. Two key regimes of parameter constellations are identified: one where full adaptation is possible, and one where adaptation requires critical regions be removed. Techniques used to derive these results include a general nonparametric minimax test for infinite-dimensional null- and alternative hypotheses, and new lower bounds for L^2-adaptive confidence sets.