DENSITY ESTIMATION BY WAVELET THRESHOLDING 1

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 coef(cid:12)cients. Minimax rates of convergence are studied over a large range of Besov function classes B (cid:27)pq and for a range of global L 0 p error measures, 1 (cid:20) 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 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 0 D 2).

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