The Annals of Statistics

A complement to Le Cam’s theorem

Mark G. Low and Harrison H. Zhou

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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

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

First available in Project Euclid: 24 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Asymptotic equivalence Poissonization decision theory additional observations


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.

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