Results 1 
8 of
8
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.
Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations
 Fields
, 1999
"... We consider adaptive estimating the value of a linear functional from indirect white noise observations. For a flexible approach, the problem is embedded in an abstract Hilbert scale. We develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collect ..."
Abstract

Cited by 27 (4 self)
 Add to MetaCart
We consider adaptive estimating the value of a linear functional from indirect white noise observations. For a flexible approach, the problem is embedded in an abstract Hilbert scale. We develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the Hilbert scale. It is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals. The case of discretized indirect white noise observations is studied, and the adaptive estimator in this setting is developed. Keywords: adaptive estimation, discretization, Hilbert scales, inverse problems, linear functionals, regularization, minimax risk. Running title: Adaptive inverse estimation of linear functionals Department of Statistics, University of Haifa, Mount Carmel, Haifa 31905, Israel. email: goldensh@rstat.haifa.ac.il y Ukrainian Academy of Sciences, Institute of Mathematics, Tereshenkivska str. 3, 252601 Kiev4, Uk...
On rate optimal local estimation in nonparametric instrumental regression.
, 2009
"... We consider the problem of estimating the value of a linear functional in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The proposed estimator is based on dimension reduction and addition ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We consider the problem of estimating the value of a linear functional in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The proposed estimator is based on dimension reduction and additional thresholding. The minimax optimal rate of convergence of the estimator is derived assuming that the structural function and the representer of the linear functional belong to some ellipsoids which are in a certain sense linked to the conditional expectation operator of Z given W. We illustrate these results by considering classical smoothness assumptions.
On rate optimal local estimation in functional linear model
, 2009
"... We consider the problem of estimating for a given representer h the value ℓh(β) of a linear functional of the slope parameter β in functional linear regression, where scalar responses Y1,..., Yn are modeled in dependence of random functions X1,..., Xn. The proposed estimators of ℓh(β) are based on d ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We consider the problem of estimating for a given representer h the value ℓh(β) of a linear functional of the slope parameter β in functional linear regression, where scalar responses Y1,..., Yn are modeled in dependence of random functions X1,..., Xn. The proposed estimators of ℓh(β) are based on dimension reduction and additional thresholding. The minimax optimal rate of convergence of the estimator is derived assuming that the slope parameter and the representer belong to some ellipsoid which are in a certain sense linked to the covariance operator associated to the regressor. We illustrate these results by considering Sobolev ellipsoids and finitely or infinitely smoothing covariance operator.
Relationship Pattern of Poverty and Unemployement in Indonesia with Bayesian Spline Approach 1
"... Poverty is one of fundamental problems which become major concern of Indonesia Government. World Poverty Commission said that unemployment is one of the main causes of poverty. A lot of literatures state that there is a strong correlation between unemployment and poverty, but to prove it empirically ..."
Abstract
 Add to MetaCart
(Show Context)
Poverty is one of fundamental problems which become major concern of Indonesia Government. World Poverty Commission said that unemployment is one of the main causes of poverty. A lot of literatures state that there is a strong correlation between unemployment and poverty, but to prove it empirically, was not easy. To see the relationship pattern between poverty and unemployment in Indonesia, it can be used spline nonparametric regression model. Spline estimator in nonparametric regression can be obtained by Bayessian approach by using prior Gaussian improper and in order to choose the optimal smoothing parameter, Generalized Cross Validation (GCV) method is choosen. Relationship model of poverty and unemployment in Indonesia obtained in the form of a quadratic spline model with two optimal knots where percentage of poverty is in quadratic curve and rise in the stage when open unemployment rate is less than 3.87, and will be declined when the open unemployment rate moved between 3.87 and 4.24. But after the open unemployment rate reached 4.24, the percentage of poverty repatterned quadratically but decreased slowly. So, for the case in Indonesia, unidirectional relationship between poverty and unemployment in the region occurred only partially, while some are actually spinning.
Minimax Estimation of Linear Functionals, Particularly in Nonparametric Regression and Positron Emission Tomography
"... Often it is required to estimate a linear functional of a function f... ..."
Assume that Y 1
"... are i.i.d. observations from a distribution with a continuous density function g. Let yE(0. A density n function estimator which can be written in the form 1 K(y,Yj) is called a kernel estimator with kernel K. Whittle (1958) proposed selecting a kernel estimator of a density, using as criterion ex ..."
Abstract
 Add to MetaCart
(Show Context)
are i.i.d. observations from a distribution with a continuous density function g. Let yE(0. A density n function estimator which can be written in the form 1 K(y,Yj) is called a kernel estimator with kernel K. Whittle (1958) proposed selecting a kernel estimator of a density, using as criterion expected square error, and observed that implementation of this approach requires the specification of only first and second moments of the joint distribution of the values of the density function g(.) at the various values of its argument. Hartigan(1969) described Whittle's approach as a "linear Bayes " approach. Brunk (1980) preferred to refer to it as "Bayesian Least Squares " because, as for ordinary least squares, both input and output involve only first and second moments. In this thesis, Brunk (1980)'s Bayesian Least Squares method has been slightly modified and applied to the estimation of univariate and mixing densities. For the univariate density estimation, we begin with a prescribed prior mean probability density g of g, and let {P r (y)} be a prescribed sequence of functions orthonormal w.r.t. g o, with P