A complement to Le Cam's theorem

TL;DR

This paper explores conditions under which density estimation experiments are asymptotically equivalent to Poisson experiments, providing a sharp Besov smoothness criterion and extending results to compact subsets of continuous functions.

Contribution

It establishes a weak assumption-based asymptotic equivalence between density estimation and Poisson experiments, with a sharp Besov smoothness condition for Poissonization.

Findings

Poissonization is possible if $oldsymbol{eta p > 1/2}$ in Besov spaces.

Poissonization fails if $oldsymbol{eta p < 1/2}$.

Asymptotic equivalence holds for compact subsets of continuous functions.

Abstract

This paper examines asymptotic equivalence in the sense of Le Cam between density estimation experiments and the accompanying Poisson experiments. The significance of asymptotic equivalence is that all asymptotically optimal statistical procedures can be carried over from one experiment to the other. The equivalence given here is established under a weak assumption on the parameter space $\mathcal{F}$. In particular, a sharp Besov smoothness condition is given on $\mathcal{F}$ which is sufficient for Poissonization, namely, if $\mathcal{F}$ is in a Besov ball $B_{p,q}^{\alpha}(M)$ with $\alpha p>1/2$. Examples show Poissonization is not possible whenever $\alpha p<1/2$. In addition, asymptotic equivalence of the density estimation model and the accompanying Poisson experiment is established for all compact subsets of $C([0,1]^m)$, a condition which includes all H\"{o}lder balls with smoothness $\alpha>0$.