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Deconvoluting kernel density estimators
 Statistics
, 1990
"... This paper considers estimation ofa continuous bounded probability density when observations from the density are contaminated by additive measurement errors having a known distribution. Properties of the estimator obtained by deconvolving a kernel estimator of the observed data are investigated. Wh ..."
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Cited by 61 (7 self)
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This paper considers estimation ofa continuous bounded probability density when observations from the density are contaminated by additive measurement errors having a known distribution. Properties of the estimator obtained by deconvolving a kernel estimator of the observed data are investigated. When the kernel used is sufficiently smooth the deconvolved estimator is shown to be pointwise consistent and bounds on its integrated mean squared error are derived. Very weak assumptions are made on the measurementerror density thereby permitting a comparison of the effects of different types of measurement error on the deconvolved estimator.
Wavelet Deconvolution
 IEEE Transactions on Information Theory
, 2002
"... This paper studies the issue of optimal deconvolution density estimation using wavelets. The approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation. We explore the asymptotic properties of estimators based on thresholding of estima ..."
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Cited by 37 (1 self)
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This paper studies the issue of optimal deconvolution density estimation using wavelets. The approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation. We explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients. Minimax rates of convergence under the integrated square loss are studied over Besov classes Bσpq of functions for both ordinary smooth and supersmooth convolution kernels. The minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels. It is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the Besov spaces with p < 2 when the convolution kernel is ordinary smooth and super smooth. If the convolution kernel is ordinary smooth, then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms. Adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed. Keywords. Adaptive estimation, Besov spaces, KullbackLeibler information, linear estimators, minimax estimation, thresholding, wavelet bases.
Sharp optimality for density deconvolution with dominating bias
 Theor. Probab. Appl
, 2005
"... bias ..."
Rates of convergence of some estimators in a class of deconvolution problems
 Statist. Probab. Letters
, 1990
"... This paper studies the problem of estimating the density of U when only independent copies of X = U + Z is observable where Z is an independent measurement error. Convergence rates of a family of deconvolved Kernel density estimators are obtained under different assumptions on the density of Z. ..."
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Cited by 15 (3 self)
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This paper studies the problem of estimating the density of U when only independent copies of X = U + Z is observable where Z is an independent measurement error. Convergence rates of a family of deconvolved Kernel density estimators are obtained under different assumptions on the density of Z.
Spherical Deconvolution
, 1998
"... This paper proposes nonparametric deconvolution density estimation over S 2 . Here we would think of the S 2 elements of interest being corrupted by random SO(3) elements (rotations). The resulting density on the observations would be a convolution of the SO(3) density with the true S 2 densit ..."
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Cited by 13 (4 self)
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This paper proposes nonparametric deconvolution density estimation over S 2 . Here we would think of the S 2 elements of interest being corrupted by random SO(3) elements (rotations). The resulting density on the observations would be a convolution of the SO(3) density with the true S 2 density. Consequently, the methodology, as in the Euclidean case, would be to use Fourier analysis on SO(3) and S 2 , involving rotational and spherical harmonics, respectively. We especially consider the case where the deconvolution operator is a bounded operator lowering the Sobolev order by a finite amount. Consistency results are obtained with rates of convergence calculated under the expected L 2 and Sobolev square norms that are proportionally inverse to some power of the sample size. As an example we introduce the rotational version of the Laplace distribution. * This research was supported in part by ARPA as administered by the AFOSR under contracts AFOSR900292 and DOD F496093105...
Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models
 Communications in Statistics, Series A
, 1989
"... generalized linear model; Hestimation; measurement error; structural IIIOdels; unbiased estimation. Let Wbe a normal random variable with mean ~ variance 0 2 • and known Conditions on the function fee) are given under which there ' exists an unbiased estimator, T(W), of f(~) real~. for all In'parti ..."
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Cited by 12 (1 self)
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generalized linear model; Hestimation; measurement error; structural IIIOdels; unbiased estimation. Let Wbe a normal random variable with mean ~ variance 0 2 • and known Conditions on the function fee) are given under which there ' exists an unbiased estimator, T(W), of f(~) real~. for all In'particular it is shown that f(e) must De an entire function over the complex plane. Infinite series solutions for fee) are obtained which are shown to be valid under growth conditions of the derivatives, f(k)(e), of fee). Approximate solutions are given for the cases in which no exact solution exists. The theory is applied to nonlinear measurementerror models as a means of finding unbiased score functions when measurement error is normally distributed. Relative efficiencies comparing the proposed method to the use of conditional scores (Stefanski and Carroll, 1987) are given for the Poisson regression model with canonical link. 1.1 The Estimation Problem 1.
Change–point estimation from indirect observations. 1
 Minimax Complexity. Ann. Inst. Henri Poincaré Probab. Stat
, 2008
"... Dedicated to Boris Polyak on the occasion of his 70th birthday We study nonparametric changepoint estimation from indirect noisy observations. Focusing on the white noise convolution model, we consider two classes of functions that are smooth apart from the changepoint. We establish lower bounds o ..."
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Cited by 8 (1 self)
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Dedicated to Boris Polyak on the occasion of his 70th birthday We study nonparametric changepoint estimation from indirect noisy observations. Focusing on the white noise convolution model, we consider two classes of functions that are smooth apart from the changepoint. We establish lower bounds on the minimax risk in estimating the changepoint and develop rate optimal estimation procedures. The results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the changepoint and by the degree of illposedness of the convolution operator. Optimality is obtained by introducing a new technique that involves, as a key element, detection of zero crossings of an estimate of the properly smoothed second derivative of the underlying function. 1. Introduction. In
Optimal ChangePoint Estimation in Inverse Problems
 Scand. J. Statist
, 1997
"... . We develop a method of estimating changepoints of a function in the case of indirect noisy observations. As two paradigmatic problems we consider deconvolution and errorsinvariables regression. We estimate the scalar products of our indirectly observed function with appropriate test functions, ..."
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Cited by 7 (0 self)
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. We develop a method of estimating changepoints of a function in the case of indirect noisy observations. As two paradigmatic problems we consider deconvolution and errorsinvariables regression. We estimate the scalar products of our indirectly observed function with appropriate test functions, which are shifted over the interval of interest. An estimator of the change point is obtained by the extremal point of this quantity. We derive rates of convergence for this estimator. They depend on the degree of illposedness of the problem, which derives from the smoothness of the error density. Analyzing the Hellinger modulus of continuity of the problem we show that these rates are minimax. 1991 Mathematics Subject Classification. Primary 62G05; secondary 62G20 Key words and phrases. Changepoint estimation, inverse problems, indirect observations, deconvolution, errorsinvariables regression, optimal rates of convergence 1 1. Introduction Changepoint estimation has often been s...