Results 1 
2 of
2
Wavelet shrinkage: asymptopia
 Journal of the Royal Statistical Society, Ser. B
, 1995
"... Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators bein ..."
Abstract

Cited by 297 (36 self)
 Add to MetaCart
Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators being obtained for a variety of interesting problems. Unfortunately, the results have often not been translated into practice, for a variety of reasons { sometimes, similarity to known methods, sometimes, computational intractability, and sometimes, lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coe cients towards the origin by an amount p p 2 log(n) = n. The method is di erent from methods in common use today, is computationally practical, and is spatially adaptive; thus it avoids a number of previous objections to minimax estimators. At the same time, the method is nearly minimax for a wide variety of loss functions { e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives { and for a wide range of smoothness classes, including standard Holder classes, Sobolev classes, and Bounded Variation. This is amuch broader nearoptimality than anything previously proposed in the minimax literature. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and informationbased complexity.
Maxisets for density estimation on R
 Math. Methods of Statist., n
, 2006
"... The problem of density estimation on R is considered. Adopting the maxiset point of view, we focus on performance of adaptive procedures. Any rule which consists in neglecting the wavelet empirical coefficients smaller than a sequence of thresholds vn will be called an elitist rule. We prove that fo ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
(Show Context)
The problem of density estimation on R is considered. Adopting the maxiset point of view, we focus on performance of adaptive procedures. Any rule which consists in neglecting the wavelet empirical coefficients smaller than a sequence of thresholds vn will be called an elitist rule. We prove that for such a procedure the maximal space for the rate v αp n, with 0 < α < 1, is always contained in the intersection of a Besov space and a weak Besov space. With no assumption on compactness of the support of the density goal f, we show that the hard thresholding rule is the best procedure among elitist rules when taking the classical choice of thresholds vn √ = µ n−1 log(n), with µ> 0. Then, we point out the significance of datadriven thresholds in density estimation by comparing the maxiset of the hard thresholding rule with the one of Juditsky and LambertLacroix’s procedure.