The Le Cam distance between density estimation, Poisson processes and Gaussian white noise

TL;DR

This paper establishes precise bounds on the Le Cam distance between density estimation, Poisson processes, and Gaussian white noise models, clarifying conditions for their asymptotic equivalence based on smoothness and density bounds.

Contribution

It provides explicit bounds and conditions for the Le Cam distance between these statistical experiments, extending understanding of their asymptotic relationships.

Findings

Derived matching lower and upper bounds for Le Cam deficiencies

Identified sharp conditions on density bounds for asymptotic equivalence

Extended results to Poisson intensity estimation

Abstract

It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities have H\"older smoothness larger than $1/2$ and are uniformly bounded away from zero. We derive matching lower and constructive upper bounds for the Le Cam deficiencies between these experiments, with explicit dependence on both the sample size and the size of the densities in the parameter space. As a consequence, we derive sharp conditions on how small the densities can be for asymptotic equivalence to hold. The related case of Poisson intensity estimation is also treated.