Density estimation by wavelet thresholding

Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic propel-ties of estimators based on thresholding of empirical wavelet coefficients. Minimax rates of convergence are studied over a large range of Besov function classes B-sigma pq and for a range of global L'(p) error measures, 1 less than or equal to p' less than or equal to infinity. 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).