Estimation of convolution in the model with noise.

Authors
  • CHESNEAU C.
  • COMTE F.
  • MABON G.
  • NAVARRO F.
Publication date
2015
Publication type
Journal Article
Summary We investigate the estimation of the $\ell$-fold convolution of the density of an unobserved variable $X$ from $n$ i.i.d. observations of the convolution model $Y=X+\varepsilon$. We first assume that the density of the noise $\varepsilon$ is known and define nonadaptive estimators, for which we provide bounds for the mean integrated squared error (MISE). In particular, under some smoothness assumptions on the densities of $X$ and $\varepsilon$, we prove that the parametric rate of convergence $1/n$ can be attained. Then we construct an adaptive estimator using a penalization approach having similar performances to the nonadaptive 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.
Publisher
Informa UK Limited
Topics of the publication
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