Asymptotic equivalence of density estimation and Gaussian white noise

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 deciency distance would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a rst result of this kind has recently been established for Gaussian regression (Brown and Low, 1993). 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 Holder ball with exponent > 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 4n 1 . This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the nite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and logf are also considered, allowing for various "automatic" asymptotic risk bounds in the i. i. d. model from white noise. As rst applications we discuss exact constants for L2 and Hellinger loss.