Data driven density estimation in presence of unknown convolution operator

Fabienne Comte (1) , Claire Lacour (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We study the following model of deconvolution $Y=X+\varepsilon$ with i.i.d. observations $Y_1,\dots, Y_n$ and $\varepsilon_{-1},\dots,\varepsilon_{-M}$. The $(X_j)_{1\leq j\leq n}$ are i.i.d. with density $f$, independent of the $\varepsilon_j$. The aim of the paper is to estimate $f$ without knowing the density $f_{\varepsilon}$ of the $\varepsilon_j$. We first define a projection estimator, for which we provide bounds for the pointwise and the integrated $L^2$-risk. We consider ordinary smooth and supersmooth noise $\varepsilon$ with regard to ordinary smooth and supersmooth densities $f$. Then we present an adaptive estimator of the density of $f$. This estimator is obtained by penalization of the projection contrast, which provides model selection. Lastly, we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints.

Mots clés: nonparametric estimation; Adaptive estimation; deconvolution; density estimation; mean square risk; minimax rates; nonparametric estimation.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Fabienne Comte, Claire Lacour. Data driven density estimation in presence of unknown convolution operator. Journal of the Royal Statistical Society: Series B, 2011, 73 (4), pp.601-627. ⟨10.1111/j.1467-9868.2011.00775.x⟩. ⟨hal-00317447v2⟩