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 deficiency distance ∆ 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 (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 Hölder ball with exponent α > 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. As first applications we discuss exact constants for L 2 and Hellinger loss.

Adaptive minimax estimation of a fractional derivative

“…Obviously, the Wicksell problem does not coincide with the problem of estimation of the fractional derivative in the Gaussian white noise model. However, they are related closely since for ε = n −1/2 in (3), on certain conditions, the corresponding statistical experiments are asymptotically equivalent in the Le Cam sense (Nussbaum 1996). In this paper we construct an adaptive asymptotically minimax estimator of the vector v(θ) under the assumption that θ belongs to the Sobolev class Θ β with unknown smoothness β.…”

Weighted angle Radon transform: Convergence rates and efficient estimation

“…where W (h) is a Gaussian field with mean zero and covariance For direct observations, regression as well as density estimation problems are asymptotically equivalent to white noise models under fairly general conditions, see Nussbaum (1996) and Reiß (2008). While no corresponding results are yet available for our indirect models, the analysis in the technically less complicated white noise model still provides valuable insight into the difficulty of the estimation problem.…”

Estimation from moments measurements for amyloid depolymerisation

“…The choice for the parameters s and p depends on the kind of noise (s = 0 for a deterministic noise, s = 1 2 and p = 2 for a deterministic equivalent of a gaussian white noise [17]). The ill-posedness of the problem comes from the fact that the noisy measurement z is in general not differentiable, so that we cannot use directly Equality (7) to solve our problem.…”