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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 ..."
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Cited by 297 (36 self)
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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.
Direct Estimation of the Index Coefficient in a SingleIndex Model,” The Annals of Statistics
, 2001
"... Singleindex modeling is widely applied in, for example, econometric studies as a compromise between too restrictive parametric models and flexible but hardly estimable purely nonparametric models. By such modeling the statistical analysis usually focuses on estimating the index coefficients. The a ..."
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Cited by 42 (5 self)
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Singleindex modeling is widely applied in, for example, econometric studies as a compromise between too restrictive parametric models and flexible but hardly estimable purely nonparametric models. By such modeling the statistical analysis usually focuses on estimating the index coefficients. The average derivative estimator (ADE) of the index vector is based on the fact that the average gradient of a single index function fxβ is proportional to the index vector β. Unfortunately, a straightforward application of this idea meets the socalled “curse of dimensionality” problem if the dimensionality d of the model is larger than 2. However, prior information about the vector β can be used for improving the quality of gradient estimation by extending the weighting kernel in a direction of small directional derivative. The method proposed in this paper consists of such iterative improvements of the original ADE. The whole procedure requires at most 2 log n iterations and the resulting estimator is nconsistent under relatively mild assumptions on the model independently of the dimensionality d.
On adaptive inverse estimation of linear functionals in Hilbert scales. Bernoulli
, 2003
"... We address the problem of estimating the value of a linear functional h f, xi from random noisy observations of y Ax in Hilbert scales. Both the white noise and density observation models are considered. We propose an estimation procedure that adapts to unknown smoothness of x, of f, and of the noi ..."
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Cited by 14 (0 self)
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We address the problem of estimating the value of a linear functional h f, xi from random noisy observations of y Ax in Hilbert scales. Both the white noise and density observation models are considered. We propose an estimation procedure that adapts to unknown smoothness of x, of f, and of the noise covariance operator. It is shown that accuracy of this adaptive estimator is worse only by a logarithmic factor than one could achieve in the case of known smoothness. As an illustrative example, the problem of deconvolving a bivariate density with singular support is considered.
ON NONPARAMETRIC TESTS OF POSITIVITY/MONOTONICITY/CONVEXITY
, 2002
"... We consider the problem of estimating the distance from an unknown signal, observed in a whitenoise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the Lrdistance, 1 ≤ r<∞, from the signal to ..."
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Cited by 4 (0 self)
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We consider the problem of estimating the distance from an unknown signal, observed in a whitenoise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the Lrdistance, 1 ≤ r<∞, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity, monotonicity and convexity of the unknown signal. We also provide an estimate for the distance to the cone of positive functions for which risk is, by a logarithmic factor, smaller than that of the “plugin ” estimate.
A Linear MiniMax Estimator for the Case of a Quartic Loss Function
, 2000
"... Let Y (t) be a stochastic process on [0; 1] modeled as dY t = `(t)dt + dW (t), where W (t) denotes a standard Wiener process, and `(t) is an unknown function assumed to belong to a given set \Theta ae L 2 [0; 1]. We consider the problem of estimating the value L(`), where L is a known continuous ..."
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Cited by 1 (1 self)
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Let Y (t) be a stochastic process on [0; 1] modeled as dY t = `(t)dt + dW (t), where W (t) denotes a standard Wiener process, and `(t) is an unknown function assumed to belong to a given set \Theta ae L 2 [0; 1]. We consider the problem of estimating the value L(`), where L is a known continuous linear function defined on \Theta, using linear estimators of the form hm; yi = R m(t)dY (t); m 2 L 2 [0; 1]. We solve the problem for the case of a quartic loss function, and compare the solution of the quartic case with the solution for the case of a quadratic loss function. 1 Introduction We consider the following linear estimation problem. Let Y (t) be a stochastic process on the interval [0,1] given by dY (t) = `(t)dt + dW (t); (1.1) where W is a standard Wiener process and ` 2 \Theta ae L 2 [0; 1]. A bounded continuous linear function L on \Theta is specified, and the statistical problem is to estimate L(`) by a linear estimator hm; Y i = R 1 0 m(t)dY (t) for the case of a quarti...
On Testing Positivity/Monotonicity/Convexity of nonparametric Signals
, 1999
"... The problem. The problem addressed in this paper is as follows: let a nonparametric signal f: [0, 1] → R be observed according to the standard “signal + white noise ” model, so that the observation is a realization of the Gaussian random process X f n(·) on [0, 1]: ..."
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The problem. The problem addressed in this paper is as follows: let a nonparametric signal f: [0, 1] → R be observed according to the standard “signal + white noise ” model, so that the observation is a realization of the Gaussian random process X f n(·) on [0, 1]:
Characterization of Linear MiniMax Estimators for Loss Functions of Arbitrary Power
"... Let Y (t), t 2 [0; 1], be a stochastic process modeled as dY t = (t)dt +dW (t), where W (t) denotes a standard Wiener process, and (t) is an unknown function assumed to belong to a given set L 2 [0; 1]. We consider the problem of estimating the value L(), where L is a continuous linear functio ..."
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Let Y (t), t 2 [0; 1], be a stochastic process modeled as dY t = (t)dt +dW (t), where W (t) denotes a standard Wiener process, and (t) is an unknown function assumed to belong to a given set L 2 [0; 1]. We consider the problem of estimating the value L(), where L is a continuous linear function dened on , using linear estimators of the form hm; yi = R m(t)dY (t); m 2 L 2 [0; 1]. The distance between the quantity L() and the estimated value is measured by a loss function. In this paper we consider the loss function to be an arbitrary even power function. We provide a characterization of the best linear minimax estimator for a general power function which implies the characterization for two special cases which have previously been considered in the literature, viz. the case of a quadratic loss function and the case of a quartic loss function. 1 Introduction We consider the following linear estimation problem. Let Y (t), t 2 [0; 1], be a stochastic process given by d...
A NOTE ON NONPARAMETRIC ESTIMATION OF LINEAR FUNCTIONALS
"... Precise asymptotic descriptions of the minimax affine risks and biasvariance tradeoffs for estimating linear functionals are given for a broad class of moduli. The results are complemented by illustrative examples including one where it is possible to construct an estimator which is fully adaptive ..."
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Precise asymptotic descriptions of the minimax affine risks and biasvariance tradeoffs for estimating linear functionals are given for a broad class of moduli. The results are complemented by illustrative examples including one where it is possible to construct an estimator which is fully adaptive over a range of parameter spaces. 1. Introduction. We