Asymptotic nonequivalence of density estimation and Gaussian white noise for small densities

TL;DR

This paper investigates the conditions under which density estimation on the unit interval is asymptotically equivalent to Gaussian white noise, revealing that a uniform lower bound on densities is generally necessary for this equivalence.

Contribution

It precisely characterizes the minimal uniform lower bound needed for asymptotic equivalence between density estimation and Gaussian white noise.

Findings

A uniform lower bound is necessary for asymptotic equivalence.

The paper sharply characterizes the size of this lower bound.

Asymptotic equivalence fails without this bound.

Abstract

It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities are sufficiently smooth and uniformly bounded away from zero. We show that a uniform lower bound, whose size we sharply characterize, is in general necessary for asymptotic equivalence to hold.