The Annals of Statistics

A complement to Le Cam’s theorem

Mark G. Low and Harrison H. Zhou

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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

Permanent link to this document
https://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

Keywords
Asymptotic equivalence Poissonization decision theory additional observations

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


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References

  • Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074--2097.
  • Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384--2398.
  • Brown, L. D. and Zhang, C.-H. (1998). Asymptotic nonequivalence of nonparametric experiments when the smoothness index is $1/2$. Ann. Statist. 26 279--287.
  • Golubev, G. K., Nussbaum, M. and Zhou, H. H. (2005). Asymptotic equivalence of spectral density estimation and Gaussian white noise. Available at www.stat.yale.edu/~hz68.
  • Grama, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167--214.
  • Johnstone, I. M. (2002). Function Estimation and Gaussian Sequence Models. Available at www-stat.stanford.edu/~imj.
  • Kolchin, V. F., Sevast'yanov, B. A. and Chistyakov, V. P. (1978). Random Allocations. Winston, Washington.
  • Le Cam, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist. 35 1419--1455.
  • Le Cam, L. (1974). On the information contained in additional observations. Ann. Statist. 4 630--649.
  • Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
  • Mammen, E. (1986). The statistical information contained in additional observations. Ann. Statist. 14 665--678.
  • Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535--543.
  • Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399--2430.
  • Woodroofe, M. (1967). On the maximum deviation of the sample density. Ann. Math. Statist. 38 475--481.
  • Yang, Y. and Barron, A. R. (1999). Information-theoretic determination of minimax rates of convergence. Ann. Statist. 27 1564--1599.