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2013 Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model
Fabien Navarro, Christophe Chesneau, Jalal Fadili, Taoufik Sassi
Electron. J. Statist. 7: 428-453 (2013). DOI: 10.1214/13-EJS776

Abstract

We observe $n$ heteroscedastic stochastic processes $\{Y_{v}(t)\}_{v}$, where for any $v\in\{1,\ldots,n\}$ and $t\in [0,1]$, $Y_{v}(t)$ is the convolution product of an unknown function $f$ and a known blurring function $g_{v}$ corrupted by Gaussian noise. Under an ordinary smoothness assumption on $g_{1},\ldots,g_{n}$, our goal is to estimate the $d$-th derivatives (in weak sense) of $f$ from the observations. We propose an adaptive estimator based on wavelet block thresholding, namely the “BlockJS estimator". Taking the mean integrated squared error (MISE), our main theoretical result investigates the minimax rates over Besov smoothness spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearly-minimax in the least favorable situation. We also report a comprehensive suite of numerical simulations to support our theoretical findings. The practical performance of our block estimator compares very favorably to existing methods of the literature on a large set of test functions.

Citation

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Fabien Navarro. Christophe Chesneau. Jalal Fadili. Taoufik Sassi. "Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model." Electron. J. Statist. 7 428 - 453, 2013. https://doi.org/10.1214/13-EJS776

Information

Published: 2013
First available in Project Euclid: 30 January 2013

zbMATH: 1337.62069
MathSciNet: MR3020428
Digital Object Identifier: 10.1214/13-EJS776

Subjects:
Primary: 62G07, 62G20
Secondary: 62F12

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society

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