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December 1996 Asymptotic equivalence of density estimation and Gaussian white noise
Michael Nussbaum
Ann. Statist. 24(6): 2399-2430 (December 1996). DOI: 10.1214/aos/1032181160


Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance $\Delta$ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression. We consider the analogous problem for the experiment given by n i.i.d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Hölder ball with exponent $\alpha > 1/2$ and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift $f^{1/2}$ and variance $1/4 n^{-1}$. This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and log f are also considered, allowing for various "automatic" asymptotic risk bounds in the i.i.d. model from white noise.


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Michael Nussbaum. "Asymptotic equivalence of density estimation and Gaussian white noise." Ann. Statist. 24 (6) 2399 - 2430, December 1996.


Published: December 1996
First available in Project Euclid: 16 September 2002

zbMATH: 0867.62035
MathSciNet: MR1425959
Digital Object Identifier: 10.1214/aos/1032181160

Primary: 62G07
Secondary: 62B15 , 62G20

Keywords: asymptotic minimax risk , Curve estimation , deficiency distance , Hungarian construction , likelihood process , Nonparametric experiments

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 6 • December 1996
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