Donoho and Johnstone (1992a) proposed a method for reconstructing an unknown function f on [0; 1] from noisy data di = f(ti)+ zi, iid i =0;:::;n 1, ti = i=n, zi N(0; 1). The reconstruction fn ^ is de ned in the wavelet domain by translating all the empirical wavelet coe cients of d towards 0 by an amount p 2 log(n) = p n. We prove two results about that estimator. [Smooth]: With high probability ^ fn is at least as smooth as f, in any of a wide variety of smoothness measures. [Adapt]: The estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. Our proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model.

We attempt to recover a function of unknown smoothness from noisy, sampled data. We introduce a procedure, SureShrink, which suppresses noise by thresholding the empirical wavelet coefficients. The thresholding is adaptive: a threshold level is assigned to each dyadic resolution level by the principle of minimizing the Stein Unbiased Estimate of Risk (Sure) for threshold estimates. The computational effort of the overall procedure is order N log(N) as a function of the sample size N. SureShrink is smoothness-adaptive: if the unknown function contains jumps, the reconstruction (essentially) does also; if the unknown function has a smooth piece, the reconstruction is (essentially) as smooth as the mother wavelet will allow. The procedure is in a sense optimally smoothness-adaptive: it is near-minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet. We know from a previous paper by the authors that traditional smoothing methods -- kernels, splines, and orthogonal series estimates -- even with optimal choices of the smoothing parameter, would be unable to perform

We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coe cients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel- and Besov-type smoothness constraints, and asymptotically minimax over Besov bodies with p q. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with p <2, so our method can signi cantly outperform every linear method (kernel, smoothing spline, sieve,:::) in a minimax sense. Variants of our method based on simple threshold nonlinearities are nearly minimax. Our method possesses the interpretation of spatial adaptivity: it reconstructs using a kernel which mayvary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper discuss practical implementation, spatial adaptation properties and applications to inverse problems.

Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nite-dimensional 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 near-optimality 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 information-based complexity.

Density estimation is a commonly used test case for non-parametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coe cients. Minimax rates of convergence are studied over a large range of Besov function classes Bs;p;q and for a range of global L 0 p error measures, 1 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 non-linearity 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 = 2).

In this paper, we present some general results determining minimax bounds on statistical risk for density estimation based on certain information-theoretic considerations. These bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence.

The paper is devoted to developing spatial adaptive estimates for restoring functions from noisy observations. We show that the traditional least square (piecewise polynomial) estimate equipped with adaptively adjusted window possesses simultaneously many attractive adaptive properties, namely, 1) it is near– optimal within ln n–factor for estimating a function (or its derivative) at a single point; 2) it is spatial adaptive in the sense that its quality is close to that one which could be achieved if smoothness of the underlying function was known in advance; 3) it is optimal in order (in the case of “strong ” accuracy measure) or near–optimal within ln n–factor (in the case of “weak ” accuracy measure) for estimating whole function (or its derivative) over wide range of the classes and global loss functions. We demonstrate that the “spatial adaptive abilities ” of our estimate are, in a sense, the best possible. Besides this, our adaptive estimate is computationally efficient and demonstrates reasonable practical behavior. 1

Pinsker(1980) gave a precise asymptotic evaluation of the minimax mean squared error of estimation of a signal in Gaussian noise when the signal is known a priori to lie in a compact ellipsoid in Hilbert space. This `Minimax Bayes ' method can be applied to a variety of global non-parametric estimation settings with parameter spaces far from ellipsoidal. For example it leads to a theory of exact asymptotic minimax estimation over norm balls in Besov and Triebel spaces using simple coordinatewise estimators and wavelet bases. This paper outlines some features of the method common to several applications. In particular, we derive new results on the exact asymptotic minimax risk over weak `p- balls in Rn as n!1, and also for a class of `local ' estimators on the Triebel scale. By its very nature, the method reveals the structure of asymptotically least favorable distributions. Thus wemaysimulate `least favorable ' sample paths. We illustrate this for estimation of a signal in Gaussian white noise over norm balls in certain Besov spaces. In wavelet bases, when p<2, the least favorable priors are sparse, and the resulting sample paths strikingly di erent from those observed in Pinsker's ellipsoidal setting (p =2).

this paper is to discuss the existence and the nature of maximal spaces in the context of nonlinear methods based on thresholding (or shrinkage) procedures. Before going further, some remarks should be made:

We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coefficients towards the origin by an amount p 2 log(n) \Delta oe= p n. 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 a broader near-optimality than anything previously proposed in the minimax literature. The theory underlying the method exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity. This paper contains a detailed proof of the result announced in Donoho, Johnstone, Kerkyacharian & Picard (1995).