A complement to Le Cam’s theorem. (English) Zbl 1194.62007

Low, Mark G.; Zhou, Harrison H.

Ann. Stat. 35, No. 3, 1146-1165 (2007).

Summary: 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 that 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 Hölder balls with smoothness \(\alpha>0\).

MSC:

62B15	Theory of statistical experiments
62G07	Density estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

asymptotic equivalence; Poissonization; decision theory; additional observations

References:

[1]	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. · Zbl 1062.62083 · doi:10.1214/009053604000000012
[2]	Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398. · Zbl 0867.62022 · doi:10.1214/aos/1032181159
[3]	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. · Zbl 0932.62061 · doi:10.1214/aos/1030563986
[4]	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. · Zbl 1181.62152
[5]	Grama, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167–214. · Zbl 0953.62039 · doi:10.1007/s004400050166
[6]	Johnstone, I. M. (2002). Function Estimation and Gaussian Sequence Models . Available at www-stat.stanford.edu/ imj.
[7]	Kolchin, V. F., Sevast’yanov, B. A. and Chistyakov, V. P. (1978). Random Allocations . Winston, Washington. · Zbl 0464.60002
[8]	Le Cam, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist. 35 1419–1455. · Zbl 0129.11202 · doi:10.1214/aoms/1177700372
[9]	Le Cam, L. (1974). On the information contained in additional observations. Ann. Statist. 4 630–649. · Zbl 0286.62004 · doi:10.1214/aos/1176342753
[10]	Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory . Springer, New York. · Zbl 0605.62002
[11]	Mammen, E. (1986). The statistical information contained in additional observations. Ann. Statist. 14 665–678. · Zbl 0633.62006 · doi:10.1214/aos/1176349945
[12]	Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535–543. · Zbl 1053.62556 · doi:10.1007/s004400050199
[13]	Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430. · Zbl 0867.62035 · doi:10.1214/aos/1032181160
[14]	Woodroofe, M. (1967). On the maximum deviation of the sample density. Ann. Math. Statist. 38 475–481. · Zbl 0157.48002 · doi:10.1214/aoms/1177698963
[15]	Yang, Y. and Barron, A. R. (1999). Information-theoretic determination of minimax rates of convergence. Ann. Statist. 27 1564–1599. · Zbl 0978.62008 · doi:10.1214/aos/1017939142