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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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