On adaptive inference and confidence bands

TL;DR

This paper investigates the conditions under which adaptive confidence bands can exist for unknown densities in Hölder classes, establishing necessary and sufficient criteria and exploring the implications for adaptive inference theory.

Contribution

It introduces a nonparametric distinguishability condition that characterizes when honest adaptive confidence bands are possible, and compares it to existing analytic conditions.

Findings

Adaptive confidence bands exist under a specific distinguishability condition.

The condition is necessary and sufficient for honest asymptotic confidence bands.

Near-optimal adaptation is achievable with standard procedures when no upper bound is known.

Abstract

The problem of existence of adaptive confidence bands for an unknown density $f$ that belongs to a nested scale of H\"{o}lder classes over $\mathbb{R}$ or $[0,1]$ is considered. Whereas honest adaptive inference in this problem is impossible already for a pair of H\"{o}lder balls $\Sigma(r),\Sigma(s),r\ne s$, of fixed radius, a nonparametric distinguishability condition is introduced under which adaptive confidence bands can be shown to exist. It is further shown that this condition is necessary and sufficient for the existence of honest asymptotic confidence bands, and that it is strictly weaker than similar analytic conditions recently employed in Gin\'{e} and Nickl [Ann. Statist. 38 (2010) 1122--1170]. The exceptional sets for which honest inference is not possible have vanishingly small probability under natural priors on H\"{o}lder balls $\Sigma(s)$. If no upper bound for the radius of the H\"{o}lder balls is known, a price for adaptation has to be paid, and near-optimal adaptation is possible for standard procedures. The implications of these findings for a general theory of adaptive inference are discussed.