Localized spherical deconvolution

Kerkyacharian, Gérard; Ngoc, Thanh Mai Pham; Picard, Dominique

doi:10.1214/10-aos858

Cited by 40 publications

(25 citation statements)

References 19 publications

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“…The problems of adaptive estimation over the scale of functional classes defined on some manifolds were studied Kerkyacharian et al (2011), Kerkyacharian et al (2012).…”

Section: Historical Notesmentioning

confidence: 99%

Adaptive estimation over anisotropic functional classes via oracle approach

Lepski¹

2015

Ann. Statist.

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We address the problem of adaptive minimax estimation in white gaussian noise model under Lp-loss, 1 ≤ p ≤ ∞, on the anisotropic Nikolskii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For proposed estimator we establish so-called Lp-norm oracle inequality and use it for deriving minimax adaptive results. We prove the existence of rate-adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular some new asymptotics of the minimax risk are discovered including necessary and sufficient conditions for existence a uniformly consistent estimator. We provide also with detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation.AMS 2000 subject classifications: 62G05, 62G20.

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“…The problems of adaptive estimation over the scale of functional classes defined on some manifolds were studied Kerkyacharian et al (2011), Kerkyacharian et al (2012).…”

Section: Historical Notesmentioning

confidence: 99%

Adaptive estimation over anisotropic functional classes via oracle approach

Lepski¹

2015

Ann. Statist.

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“…Actually, this is a natural method that is used by practitioners or theoreticians. See for example Kerkyacharian et al. (2010) who studied spherical deconvolution and replaced the characteristic function of the noise by its empirical version, since the noise is unknown in the context of astrophysics.…”

Section: Introductionmentioning

confidence: 99%

“…Actually, this is a natural method that is used by practitioners or theoreticians. See for example Kerkyacharian et al (2010) who studied spherical deconvolution and replaced the characteristic function of the noise by its empirical version, since the noise is unknown in the context of astrophysics. One of the most typical domains where a preliminary estimation of the measurement error is done is spectrometry, or spectrofluorimetry, but let us detail an example in microscopy.…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Density Estimation in the Presence of Additive Noise with unknown Distribution

Comte

Lacour

2011

Journal of the Royal Statistical Society Series B: Statistical Methodology

126

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We study the model Y D X C ". We assume that we have at our disposal independent identically distributed observations Y 1 ,...,Y n and " 1 ,...," M . The .X j / 1 j n are independent identically distributed with density f , independent of the ." j / 1 j n , independent identically distributed with density f " . The aim of the paper is to estimate f without knowing f " . We first define an estimator, for which we provide bounds for the integrated L 2 -risk.We consider ordinary smooth and supersmooth noise " with regard to ordinary smooth and supersmooth densities f. Then we present an adaptive estimator of the density of f. This estimator is obtained by penalization of a projection contrast and yields to model selection. Lastly, we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints.

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“…We only mention here some of the statistical challenges posed by astrophysical data: denoising of signals, testing stationarity, rotation invariance or gaussianity of signals, investigating the fundamental properties of the cosmic microwave background (CMB), impainting of the CMB in zones on the sphere obstructed by other radiations, producing cosmological maps, exploring clusters of galaxies or point sources, investigating the true nature of ultra high energy cosmic rays (UHECR). We refer the reader to the overview by Starck, Murtagh, and Fadili [27] of the use of various wavelet tools in this domain as well as the work of some of the authors in this direction [1] and [22].…”

Section: Introductionmentioning

confidence: 99%

Kernel and wavelet density estimators on manifolds and more general metric spaces

Cleanthous¹,

Georgiadis²,

Kerkyacharian³

et al. 2020

Bernoulli

Self Cite

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We consider the problem of estimating the density of observations taking values in classical or nonclassical spaces such as manifolds and more general metric spaces. Our setting is quite general but also sufficiently rich in allowing the development of smooth functional calculus with well localized spectral kernels, Besov regularity spaces, and wavelet type systems. Kernel and both linear and nonlinear wavelet density estimators are introduced and studied. Convergence rates for these estimators are established, which are analogous to the existing results in the classical setting of real-valued variables.

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Localized spherical deconvolution

Cited by 40 publications

References 19 publications

Adaptive estimation over anisotropic functional classes via oracle approach

Adaptive estimation over anisotropic functional classes via oracle approach

Data-Driven Density Estimation in the Presence of Additive Noise with unknown Distribution

Kernel and wavelet density estimators on manifolds and more general metric spaces

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