## The Annals of Statistics

### A complement to Le Cam’s theorem

#### 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 ℱ. In particular, a sharp Besov smoothness condition is given on ℱ which is sufficient for Poissonization, namely, if ℱ is in a Besov ball Bp,qα(M) with αp>1/2. Examples show Poissonization is not possible whenever αp<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 Hö lder balls with smoothness α>0.

#### Article information

Source
Ann. Statist. Volume 35, Number 3 (2007), 1146-1165.

Dates
First available in Project Euclid: 24 July 2007

http://projecteuclid.org/euclid.aos/1185304001

Digital Object Identifier
doi:10.1214/009053607000000091

Mathematical Reviews number (MathSciNet)
MR2341701

Zentralblatt MATH identifier
1194.62007

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62G08: Nonparametric regression

#### Citation

Low, Mark G.; Zhou, Harrison H. A complement to Le Cam’s theorem. Ann. Statist. 35 (2007), no. 3, 1146--1165. doi:10.1214/009053607000000091. http://projecteuclid.org/euclid.aos/1185304001.

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