## The Annals of Statistics

### Asymptotic equivalence of density estimation and Gaussian white noise

Michael Nussbaum

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 24, Number 6 (1996), 2399-2430.

Dates
First available in Project Euclid: 16 September 2002

https://projecteuclid.org/euclid.aos/1032181160

Digital Object Identifier
doi:10.1214/aos/1032181160

Mathematical Reviews number (MathSciNet)
MR1425959

Zentralblatt MATH identifier
0867.62035

Subjects
Primary: 62G07: Density estimation
Secondary: 62B15: Theory of statistical experiments 62G20: Asymptotic properties

#### Citation

Nussbaum, Michael. Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 (1996), no. 6, 2399--2430. doi:10.1214/aos/1032181160. https://projecteuclid.org/euclid.aos/1032181160

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