Results 1  10
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16
Sharp Adaptation for Inverse Problems With Random Noise
, 2000
"... We consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems, in particular multivariate ones. We propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellip ..."
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Cited by 74 (8 self)
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We consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems, in particular multivariate ones. We propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale (that includes ellipoids with polynomially and exponentially decreasing axes) and, at the same time, satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone nondecreasing weights. As application, we construct sharp adaptive estimators in the problems of deconvolution and tomography.
Oracle Inequalities for Inverse Problems
, 2000
"... We consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations. Let a finite set of linear estimators be given. Our aim is to mimic the estimator in that has the smallest risk on the true f . Under general conditions, ..."
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Cited by 64 (8 self)
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We consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations. Let a finite set of linear estimators be given. Our aim is to mimic the estimator in that has the smallest risk on the true f . Under general conditions, we show that this can be achieved by simple minimization of unbiased risk estimator, provided the singular values of the operator of the inverse problem decrease as a power law. The main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact. This inequality can be also used to obtain sharp minimax adaptive results. In particular, we apply it to show that minimax adaptation on ellipsoids in multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures. Mathematics Subject Classifications: 62G05, 62G20 Key Words: Statistical inverse problems, Oracle inequaliti...
Linear and convex aggregation of density estimators
, 2004
"... We study the problem of learning the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lowe ..."
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Cited by 30 (1 self)
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We study the problem of learning the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lower bounds showing that these procedures attain optimal rates of aggregation. As an example, we consider aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense. We prove that, for Pinsker’s kernel, the proposed aggregates are sharp asymptotically minimax simultaneously over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex aggregation procedure.
General empirical Bayes wavelet methods and exactly adaptive minimax estimation

, 2005
"... In many statistical problems, stochastic signals can be represented as a sequence of noisy wavelet coefficients. In this paper, we develop general empirical Bayes methods for the estimation of true signal. Our estimators approximate certain oracle separable rules and achieve adaptation to ideal risk ..."
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Cited by 26 (3 self)
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In many statistical problems, stochastic signals can be represented as a sequence of noisy wavelet coefficients. In this paper, we develop general empirical Bayes methods for the estimation of true signal. Our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals. In particular, our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in Besov balls of all smoothness and shape indices, and they are uniformly superefficient in convergence rates in all compact sets in Besov spaces with a finite secondary shape parameter. Furthermore, in classes nested between Besov balls of the same smoothness index, our estimators dominate threshold and James–Stein estimators within an infinitesimal fraction of the minimax risks. More general block empirical Bayes estimators are developed. Both white noise with drift and nonparametric regression are considered.
On minimax density estimation on R
 Bernoulli
, 2004
"... Abstract: the problem of density estimation on R from an independent sample X1,...XN with common density f is concerned. The behavior of the minimax Lprisk, 1 ≤ p ≤ ∞, is studied when f belongs to a Hölder class of regularity s on the real line. The lower bound for the minimax risk is provided. We ..."
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Cited by 21 (0 self)
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Abstract: the problem of density estimation on R from an independent sample X1,...XN with common density f is concerned. The behavior of the minimax Lprisk, 1 ≤ p ≤ ∞, is studied when f belongs to a Hölder class of regularity s on the real line. The lower bound for the minimax risk is provided. We show that the linear estimator is not efficient in this setting and construct a wavelet adaptive estimator which attains (up to a logarithmic factor in N) the lower bounds involved. We show that the minimax risk depends on the parameter p when p < 2 + 1 s. Key words: nonparametric density estimation, minimax estimation, adaptive estimation. 1
Sparse density estimation with ℓ1 penalties
 In Proceedings of 20th Annual Conference on Learning Theory (COLT 2007) (2007
"... Abstract. This paper studies oracle properties of ℓ1penalized estimators of a probability density. We show that the penalized least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in terms of the number of nonzero components of the oracle vector. The results are valid even w ..."
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Cited by 9 (2 self)
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Abstract. This paper studies oracle properties of ℓ1penalized estimators of a probability density. We show that the penalized least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in terms of the number of nonzero components of the oracle vector. The results are valid even when the dimension of the model is (much) larger than the sample size. They are applied to estimation in sparse highdimensional mixture models, to nonparametric adaptive density estimation and to the problem of aggregation of density estimators. 1
Adaptive estimation of and oracle inequalities for probability densities
, 2004
"... The theory of adaptive estimation and oracle inequalities for the case of Gaussianshift–finiteinterval experiments has made significant progress in recent years. In particular, sharpminimax adaptive estimators and exact exponentialtype oracle inequalities have been suggested for a vast set of fu ..."
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Cited by 6 (1 self)
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The theory of adaptive estimation and oracle inequalities for the case of Gaussianshift–finiteinterval experiments has made significant progress in recent years. In particular, sharpminimax adaptive estimators and exact exponentialtype oracle inequalities have been suggested for a vast set of functions including analytic and Sobolev with any positive index as well as for Efromovich–Pinsker and Stein blockwiseshrinkage estimators. Is it possible to obtain similar results for a more interesting applied problem of density estimation and/or the dual problem of characteristic function estimation? The answer is “yes. ” In particular, the obtained results include exact exponentialtype oracle inequalities which allow to consider, for the first time in the literature, a simultaneous sharpminimax estimation of Sobolev densities with any positive index (not necessarily larger than 1/2), infinitely differentiable densities (including analytic, entire and stable), as well as of not absolutely integrable characteristic functions. The same adaptive estimator is also rate minimax over a familiar class of distributions with bounded spectrum where the density and the characteristic function can be estimated with the parametric rate. 1. Introduction. Univariate
Sparse density estimation with `1 penalties
"... Abstract. This paper studies oracle properties of `1penalized estimators of a probability density. We show that the penalized least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in terms of the number of nonzero components of the oracle vector. The results are valid even ..."
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Cited by 4 (0 self)
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Abstract. This paper studies oracle properties of `1penalized estimators of a probability density. We show that the penalized least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in terms of the number of nonzero components of the oracle vector. The results are valid even when the dimension of the model is (much) larger than the sample size. They are applied to estimation in sparse highdimensional mixture models, to nonparametric adaptive density estimation and to the problem of aggregation of density estimators. 1
EXACT MINIMAX RISK FOR DENSITY ESTIMATORS IN NONINTEGER SOBOLEV CLASSES
, 2008
"... The L2minimax risk in Sobolev classes of densities with noninteger smoothness index is shown to have an analog form to that in integer Sobolev classes. To this end, the notion of Sobolev classes is generalized to fractional derivatives of order β ∈ R +. A minimax kernel density estimator for such ..."
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Cited by 2 (0 self)
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The L2minimax risk in Sobolev classes of densities with noninteger smoothness index is shown to have an analog form to that in integer Sobolev classes. To this end, the notion of Sobolev classes is generalized to fractional derivatives of order β ∈ R +. A minimax kernel density estimator for such a classes is found. Although there exists no corresponding proof in the literature so far, the result of this article was used implicitly in numerous papers. A certain necessity that this gap had to be filled, can thus not be denied.