Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model

Fabien Navarro 1, 2 Christophe Chesneau 1 Jalal M. Fadili 2 Taoufik Sassi 1
2 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
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.
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Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2013, 7, pp.428-453. 〈10.1214/13-EJS776〉
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Fabien Navarro, Christophe Chesneau, Jalal M. Fadili, Taoufik Sassi. Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2013, 7, pp.428-453. 〈10.1214/13-EJS776〉. 〈hal-00661544v3〉

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