## Electronic Journal of Statistics

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

#### 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.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 428-453.

Dates
First available in Project Euclid: 30 January 2013

https://projecteuclid.org/euclid.ejs/1359564357

Digital Object Identifier
doi:10.1214/13-EJS776

Mathematical Reviews number (MathSciNet)
MR3020428

Zentralblatt MATH identifier
1337.62069

Subjects
Primary: 62G07: Density estimation 62G20: Asymptotic properties
Secondary: 62F12: Asymptotic properties of estimators

#### Citation

Navarro, Fabien; Chesneau, Christophe; Fadili, Jalal; Sassi, Taoufik. Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model. Electron. J. Statist. 7 (2013), 428--453. doi:10.1214/13-EJS776. https://projecteuclid.org/euclid.ejs/1359564357

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