Asymptotic equivalence of density estimation and white noise.
- Nussbaum, Michael
2010 Mathematics Subject Classification
- 62G07 62B15 62G20
- Nonparametric experiments, deficiency distance, curve estimation, likelihood ratio process, Hungarian construction, asymptotic minimax risk, exact constants, Hellinger loss, linear wavelets estimators
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.