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Asymptotic equivalence of density estimation and Gaussian white noise. (English) Zbl 0867.62035
Summary: 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 $$j^{1/2}$$ and variance $$(4n)^{-1}$$. This represents a nonparametric analog of Le Cam’s heteroscedastic Gaussian approximation [L. Le Cam, Ann. Inst. H. Poincaré 21, 225-287 (1985; Zbl 0584.62024)] 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.

##### MSC:
 62G07 Density estimation 62B15 Theory of statistical experiments 62M99 Inference from stochastic processes 62G20 Asymptotic properties of nonparametric inference
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