Wavelet-based estimation of the derivatives of a function from a heteroscedastic multichannel convolution model

Fabien Navarro (1) , Christophe Chesneau (1) , Jalal M. Fadili (2) , Taoufik Sassi (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 Equipe Image - Laboratoire GREYC - UMR6072

We observe $n$ heteroscedastic stochastic processes where, for any $v\in\{1,\ldots,n\}$, a convolution product of an unknown function $f$ and a known function $g_v$ is corrupted by Gaussian noise. Under a particular ordinary smooth assumption on $g_1,\ldots,g_n$, we aim to estimate the $d$-th derivatives of $f$ from the observations. We consider an adaptive estimator based on a particular wavelet block thresholding: the "BlockJS estimator". Taking the mean integrated squared error (MISE), we prove that it achieves near optimal rates of convergence over a wide range of smoothness classes. The theory is illustrated with some numerical examples. Performance comparisons with some others methods existing in the literature are provided.

Mots clés: deconvolution; derivatives function estimation; wavelets; block thresholding; minimax

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Fabien Navarro, Christophe Chesneau, Jalal M. Fadili, Taoufik Sassi. Wavelet-based estimation of the derivatives of a function from a heteroscedastic multichannel convolution model. 2012. ⟨hal-00661544v1⟩