Sobolev-Hermite versus Sobolev nonparametric density estimation on R

Denis Belomestny (1) , Fabienne Comte (2) , Valentine Genon-Catalot (2)

1 Universität Duisburg-Essen = University of Duisburg-Essen [Essen]
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

In this paper, our aim is to revisit the nonparametric estimation of f assuming that f is square integrable on R, by using projection estimators on a Hermite basis. These estimators are defined and studied from the point of view of their mean integrated squared error on R. A model selection method is described and proved to perform an automatic bias variance compromise. Then, we present another collection of estimators, of deconvolution type, for which we define another model selection strategy. Considering Sobolev and Sobolev-Hermite spaces, the asymptotic rates of these estimators can be computed and compared: they are mainly proved to be equivalent. However, complexity evaluations prove that the Hermite estimators have a much lower computational cost than their deconvolution (or kernel) counterparts. These results are illustrated through a small simulation study.

Domains: Statistics [math.ST]

Denis Belomestny, Fabienne Comte, Valentine Genon-Catalot. Sobolev-Hermite versus Sobolev nonparametric density estimation on R. Annals of the Institute of Statistical Mathematics, 2019, ⟨10.1007/s10463-017-0624-y⟩. ⟨hal-01372985⟩