Article
LogDensity Deconvolution by Wavelet Thresholding
Institut d'�conomie Industrielle (IDEI), Toulouse, IDEI Working Papers 01/2009;
Source: RePEc
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Article: Wavelet deconvolution
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ABSTRACT: 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 B<sub>σpq</sub> 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 discussedIEEE Transactions on Information Theory 04/2002; · 2.62 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The analysis of the numerical aspects of Hilbert transform spectroscopy based on the a.c. Josephson effect is presented. The resolving power of Hilbert transform spectroscopy is determined by such factors as the linewidth of the Josephson oscillations (intrinsic or natural resolution limit) and the limitation of the measurement interval (extrinsic or technical resolution limit) like in any spectroscopic technique based on some integral transformations. The deconvolution problem in Hilbert transform spectroscopy is posed and its solution is considered using the approach of the 1st kind integral equation for the spectrum of the incident radiation constructed from the input data of the Hilbert transform spectroscopythe `hilbertogram'. The program package RECOVERY based on the maximum likelihood method is used for this purpose. This method allows to attain the maximum possible resolution enhancement in output result for a given signaltonoise ratio in the input experimental data. The samples of numerical simulations and the spectrum of frequencymodulated BWO radiation measured by means of the Josephson junction made from highTc superconductor are presented. It is shown also that the integral equation approach allows to recover the sought spectrum beyond the intrinsic resolution limit and to achieve the superresolutionComputer Physics Communications 01/2003; 151(2):171. · 2.41 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: . In this paper we consider logspline density estimation for random variables which are contaminated with random noise. In the logspline density estimation for data without noise, the logarithm of an unknown density function is estimated by a polynomial spline, the unknown parameters of which are given by maximum likelihood. When noise is present, Bsplines and the Fourier inversion formula are used to construct the logspline density estimator of the unknown density function. Rates of convergence are established when the logdensity function is assumed to be in a Besov space. It is shown that convergence rates depend on the smoothness of the density function and the decay rate of the characteristic function of the noise. Simulated data are used to show the finitesample performance of inference based on the logspline density estimation.Scandinavian Journal of Statistics 02/1999; 26(1):7386. · 1.17 Impact Factor
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