Estimation of convolution in the model with noise

Abstract

We investigate the estimation of the ℓ-fold convolution of the density of an unobserved variable X from n i.i.d. observations of the convolution model . We first assume that the density of the noise ϵ is known and define non-adaptive estimators, for which we provide bounds for the mean integrated squared error. In particular, under some smoothness assumptions on the densities of X and ϵ, we prove that the parametric rate of convergence can be attained. Then, we construct an adaptive estimator using a penalisation approach having similar performances to the non-adaptive one. The price for its adaptivity is a logarithmic term. The results are extended to the case of unknown noise density, under the condition that an independent noise sample is available. Lastly, we report a simulation study to support our theoretical findings.

Keywords:

• adaptive estimation
• convolution of densities
• measurement errors
• oracle inequality
• nonparametric estimator

AMS Subject Classifications:

• 62G07
• 62G05

Acknowledgments

The authors are thankful to the Associate Editor and anonymous reviewers for insightful suggestions, which greatly improved this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

C. Chesneau http://orcid.org/0000-0002-1522-9292