Electronic Journal of Statistics

Adaptivity in convolution models with partially known noise distribution

Cristina Butucea, Catherine Matias, and Christophe Pouet

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We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density f and some partially known noise density g. In this work, g is assumed exponentially smooth with stable law having unknown self-similarity index s. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities f, with smoothness parameter β. In this context, we first provide a consistent estimation procedure for s. This estimator is then plugged-into three different procedures: estimation of the unknown density f, of the functional f2 and goodness-of-fit test of the hypothesis H0:f=f0, where the alternative H1 is expressed with respect to $\mathbb{L}_{2}$-norm (i.e. has the form $\psi_{n}^{-2}\|f-f_{0}\|_{2}^{2}\ge \mathcal{C}$). These procedures are adaptive with respect to both s and β and attain the rates which are known optimal for known values of s and β. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of s is illustrated on synthetic data.

Article information

Electron. J. Statist. Volume 2 (2008), 897-915.

First available in Project Euclid: 3 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62G05: Estimation
Secondary: 62G10: Hypothesis testing 62G20: Asymptotic properties

Adaptive nonparametric tests convolution model goodness-of-fit tests infinitely differentiable functions partially known noise quadratic functional estimation Sobolev classes stable laws


Butucea, Cristina; Matias, Catherine; Pouet, Christophe. Adaptivity in convolution models with partially known noise distribution. Electron. J. Statist. 2 (2008), 897--915. doi:10.1214/08-EJS225. http://projecteuclid.org/euclid.ejs/1223057406.

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