Asymptotic equivalence of density estimation and white noise.

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 purified form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance Δ would make it precise. Then two models are 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 (Brown and Low, 1992). We consider the analogous problem for the experiment given by n i. i. d. observations having density ƒ on the unit interval. Our basic result concerns the parameter space of densities which are in a Sobolev class of order 4 and uniformly bounded away from zero. We show that an i. i. d. sample of size n with density ƒ is globally asymptotically equivalent to a white noise experiment with trend ƒ1/2 and variance 1⁄4n-1. This represents a nonparametric analog of Le Cam's heteroskedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques, especially the Hungarian construction. White noise models on ƒ and log ƒ are also considered, allowing for various "automatic" asymptotic risk bounds in the i. i. d. model from white noise. As first applications we discuss linear wavelet estimators of a density and exact constants for Hellinger loss