We attempt to recover an n-dimensional vector observed in white noise, where n is large and the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing power-law decay bounds on the ordered entries; and controlling the ℓ p norm for p small. We obtain a procedure which is asymptotically minimax for ℓ r loss, simultaneously throughout a range of such sparsity classes.The optimal procedure is a data-adaptive thresholding scheme, driven by control of the False Discovery Rate (FDR). FDR control is a relatively recent innovation in simultaneous testing, ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections.In our treatment, the FDR control parameter q n also plays a determining role in asymptotic minimaxity. If q = lim q n ∈ [0, 1/2] and also q n > γ/ log(n) we get sharp asymptotic minimaxity, simultaneously, over a wide range of sparse parameter spaces and loss functions. On the other hand, q = lim q n ∈ (1/2, 1], forces the risk to exceed the minimax risk by a factor growing with q.To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new.Our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors. These new rules impose complexity penalization of the form 2 · log( potential model size / actual model size ). We exhibit a close connection with FDR-controlling procedures under stringent control of the false discovery rate.Acknowledgements: ABDJ would like to acknowledge the support of Israel-USA BSF grant 1999441. IMJ would like to acknowledge support of NSF grants dms 95-05151, 00-77621, an NIH grant, a Guggenheim Foundation Fellowship, and the Australian National University. DLD would like to thank FA and YB for hospitality at Tel Aviv University during a sabbatical there, and would like to acknowledge the support of AFOSR MURI 95-P49620-96-1-0028.R n (m p [η n ]) ∼ n ρ S (τ η , µ) π 1 (dµ).

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coef®cients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior speci®ed, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any speci®c Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coef®cients. However, prior knowledge about a function's regularity properties might be dif®cult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.

A wide variety of scienti c settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise. Primary interest is in some function f(t) but the data is accessible only about some transform (Kf)(t), where K is some linear operator, and (Kf)(t) is in addition corrupted by noise. The usual linear methods for such inverse problems, for example those based on singular value decompositions, do not perform satisfactorily when the original function f(t) is spatially inhomogeneous. One alternative that has been suggested is the wavelet{ vaguelette decomposition method, based on the expansion of the unknown f(t) in wavelet series. The vaguelette{wavelet decomposition method proposed in this paper is also based on wavelet expansion. In contrast to wavelet{vaguelette decomposition, the observed data are expanded directly in wavelet series. Using exact risk calculations, the performances of the two wavelet-based methods are compared with one another and with singular value decomposition methods, in the context of the estimation of the derivative of a function observed subject to noise. A result is proved demonstrating that, with a suitable universal threshold somewhat larger than that used for standard denoising problems, both wavelet-based approaches have an ideal spatial adaptivity property.

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In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this paper gives a relatively accessible introduction to standard wavelet analysis and provides a review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be effective, rather than to researchers who are already familiar with the ®eld. Given that objective, we do not emphasize mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in greater detail and generality if required. The paper ®rst establishes some necessary basic mathematical background and terminology relating to wavelets. It then reviews the more well-established applications of wavelets in statistics including their use in nonparametric regression, density estimation, inverse problems, changepoint problems and in some specialized aspects of time series analysis. Possible extensions to the uses of wavelets in statistics are then considered. The paper concludes with a brief reference to readily available software packages for wavelet analysis.