Abstract

Density estimation is a commonly used test case for non-parametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coe cients. Minimax rates of convergence are studied over a large range of Besov function classes Bs;p;q and for a range of global L 0 p error measures, 1 p 0 < 1. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when p 0> p, some form of non-linearity is essential, since the minimax linear estimators are suboptimal by polynomial powers of n. A second approach, using an approximation of a Gaussian white noise model in a Mallows metric, is used to attain exactly optimal rates of convergence for quadratic error (p 0 = 2).

Citations