Laguerre deconvolution with unknown matrix operator

Fabienne Comte (1) , Gwennaëlle Mabon (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 Institut für Mathematik [Humboldt]

In this paper we consider the convolution model Z = X + Y with X of unknown density f , independent of Y , when both random variables are nonnegative. Our goal is to estimate the unknown density f of X from n independent, identically, distributed observations of Z, when the law of the additive process Y is unknown. When the density of Y is known, a solution to the problem has been proposed in Mabon (2016b). To make the problem identifiable for unknown density of Y , we assume that we have access to a preliminary sample of the nuisance process Y. The question is to propose a solution to an inverse problem with unknown operator. To that aim, we build a family of projection estimators of f on the Laguerre basis, particularly adapted to the non-negativeness of both random variables. The dimension of the projection space is chosen thanks to a model selection procedure by penalization. At last we prove that the final estimator satisfies an oracle inequality. It can be noted that the study of the mean integrated square risk is based on Bernstein's type concentration inequalities developed for random matrices in Tropp (2015). Finally we illustrate our method on some simulated data.

Mots clés: Non negative random variables; Linear inverse problem; Convolution model; Oracle inequalities; Adaptive estimation; Laguerre basis; Nonparametric density estimation; Random matrix

Domaines: Statistiques [math.ST]

Fabienne Comte, Gwennaëlle Mabon. Laguerre deconvolution with unknown matrix operator. 2016. ⟨hal-01416412⟩