Density estimation by wavelet thresholding

Abstract

Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients. Minimax rates of convergence are studied over a large range of Besov function classes Bσpq and for a range of global L′p error measures, 1 ≤ p′ < ∞. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when p′ > p, some form of nonlinearity 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′ = 2).