2009
DOI: 10.1214/08-aos652
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Deconvolution with unknown error distribution

Abstract: We consider the problem of estimating a density $f_X$ using a sample $Y_1,...,Y_n$ from $f_Y=f_X\star f_{\epsilon}$, where $f_{\epsilon}$ is an unknown density. We assume that an additional sample $\epsilon_1,...,\epsilon_m$ from $f_{\epsilon}$ is observed. Estimators of $f_X$ and its derivatives are constructed by using nonparametric estimators of $f_Y$ and $f_{\epsilon}$ and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknow… Show more

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Cited by 76 publications
(72 citation statements)
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“…For instance, in a physical context, a preliminary sample of the noise can be obtained. Neumann (1997) first proposed an estimation strategy still based on Fourier inversion; for the study of convergence rates, see Neumann (1997), Johannes (2009) or Meister (2009). The rigorous study of adaptive procedures in a deconvolution model with unknown errors has only recently been addressed.…”
Section: Bibliography For Real-valued Variablesmentioning
confidence: 99%
“…For instance, in a physical context, a preliminary sample of the noise can be obtained. Neumann (1997) first proposed an estimation strategy still based on Fourier inversion; for the study of convergence rates, see Neumann (1997), Johannes (2009) or Meister (2009). The rigorous study of adaptive procedures in a deconvolution model with unknown errors has only recently been addressed.…”
Section: Bibliography For Real-valued Variablesmentioning
confidence: 99%
“…In the first papers, the noise distribution is assumed to be known, see [29] for nonadaptive kernel, [57] for adaptive wavelet estimator and [22] for adaptive cut-off selection. More recently, the case of unknown noise distribution has been considered, see [55], [41], [21], [44]. The estimation of the Lévy density for Lévy processes relies on the explicit form of the characteristic function and thus takes inspiration in the deconvolution methods.…”
Section: Bibliographic Commentsmentioning
confidence: 99%
“…Several papers focus on that matter as those of Comte and Lacour (2011), Johannes and Schwarz (2013), Dattner et al (2013), Kappus (2014) and Kappus and Mabon (2014). Rates of convergence have been presented in Neumann (1997) and, more recently, in Johannes (2009), or Meister (2009 under the assumption that a preliminary sample of the noise ε is observed.…”
Section: Introductionmentioning
confidence: 99%