Open Access
2008 Adaptivity in convolution models with partially known noise distribution
Cristina Butucea, Catherine Matias, Christophe Pouet
Electron. J. Statist. 2: 897-915 (2008). DOI: 10.1214/08-EJS225


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.


Download Citation

Cristina Butucea. Catherine Matias. Christophe Pouet. "Adaptivity in convolution models with partially known noise distribution." Electron. J. Statist. 2 897 - 915, 2008.


Published: 2008
First available in Project Euclid: 3 October 2008

zbMATH: 1320.62066
MathSciNet: MR2447344
Digital Object Identifier: 10.1214/08-EJS225

Primary: 62F12 , 62G05
Secondary: 62G10 , 62G20

Keywords: Adaptive nonparametric tests , Convolution model , Goodness-of-fit tests , Infinitely differentiable functions , Partially known noise , Quadratic functional estimation , Sobolev classes , Stable laws

Rights: Copyright © 2008 The Institute of Mathematical Statistics and the Bernoulli Society

Back to Top