Asymptotic nonequivalence of density estimation and Gaussian white noise for small densities

Abstract

It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities are sufficiently smooth and uniformly bounded away from zero. We show that a uniform lower bound, whose size we sharply characterize, is in general necessary for asymptotic equivalence to hold.

20 pages, 1 figure. Some results from an early version of arXiv:1608.01824 are now found here