Title
Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates
Abbreviated Journal Title
Electron. J. Stat.
Keywords
Functional deconvolution; minimax convergence rate; hyperbolic wavelets; seismic inversion; BOXCAR DECONVOLUTION; WAVELET DECONVOLUTION; NONLINEAR ESTIMATION; INVERSE PROBLEMS; DECOMPOSITION; Statistics & Probability
Abstract
In the present paper we consider the problem of estimating a periodic (r + 1)-dimensional function f based on observations from its noisy convolution. We construct a wavelet estimator of f , derive minimax lower bounds for the L-2-risk when f belongs to a Besov ball of mixed smoothness and demonstrate that the wavelet estimator is adaptive and asymptotically near-optimal within a logarithmic factor, in a wide range of Besov balls. We prove in particular that choosing this type of mixed smoothness leads to rates of convergence which are free of the curse of dimensionality and, hence, are higher than usual convergence rates when r is large. The problem studied in the paper is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Indeed, we show that unless the function f is very smooth in the direction of the profiles, very spatially inhomogeneous along the other direction and the number of profiles is very limited, the functional deconvolution solution has a much better precision compared to a combination of M solutions of separate convolution equations. A limited simulation study in the case of r = 1 confirms theoretical claims of the paper.
Journal Title
Electronic Journal of Statistics
Volume
7
Publication Date
1-1-2013
Document Type
Article
DOI Link
Language
English
First Page
1686
Last Page
1715
WOS Identifier
ISSN
1935-7524
Recommended Citation
Benhaddou, Rida; Pensky, Marianna; and Picard, Dominique, "Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates" (2013). Faculty Bibliography 2010s. 3696.
https://stars.library.ucf.edu/facultybib2010/3696
Comments
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