A complement to Le Cam's theorem

Details A complement to Le Cam's theorem A complement to Le Cam's theorem

2007-08-16

arxiv.org/abs/0708.2233v1

Annals of Statistics 2007, Vol. 35, No. 3, 1146-1165

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053607000000091 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053607000000091

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 $\alpha>0$.

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