DENSITY-ESTIMATION BY WAVELET THRESHOLDING

Density estimation is a commonly used test case for nonparametric esti mation methods. We explore the asymptotic propel-ties of estimators ba sed on thresholding of empirical wavelet coefficients. Minimax rates o f 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 tha n or equal to p' less than or equal to infinity. A single wavelet thre shold estimator is asymptotically minimax within logarithmic terms sim ultaneously 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).