Hermite density deconvolution

We consider the additive model: Z = X + e, where X and e are independent. We construct a new estimator of the density of X from n observations of Z. We propose a projection method which exploits the specific properties of the Hermite basis. We study the quality of the resulting estimator by proving a bound on the integrated quadratic risk. We then propose an adaptive estimation procedure, that is a method of selecting a relevant model. We check that our estimator reaches the classical convergence speeds of deconvolution. Numerical simulations are proposed and a comparison with the results of the method proposed in Comte and Lacour (2011) is performed.

[1]  F. Comte,et al.  Sobolev-Hermite versus Sobolev nonparametric density estimation on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ , 2017, Annals of the Institute of Statistical Mathematics.

[2]  M. Talagrand.  New concentration inequalities in product spaces , 1996 .

[3]  F. Comte,et al.  Adaptive density deconvolution with dependent inputs , 2006, math/0606166.

[4]  G. Viennet.  Inequalities for absolutely regular sequences: application to density estimation , 1997 .

[5]  M. Ledoux.  On Talagrand's deviation inequalities for product measures , 1997 .

[6]  Claire Lacour,et al.  Data driven density estimation in presence of unknown convolution operator , 2011 .

[7]  Claire Lacour,et al.  Data‐driven density estimation in the presence of additive noise with unknown distribution , 2011 .

[8]  Jianqing Fan,et al.  Adaptively Local One-Dimensional Subproblems with Application to a Deconvolution Problem , 1993 .

[9]  V. Genon-Catalot,et al.  Laguerre and Hermite bases for inverse problems , 2018, Journal of the Korean Statistical Society.

[10]  E. Rio,et al.  Concentration around the mean for maxima of empirical processes , 2005, math/0506594.

[11]  Johanna Kappus,et al.  Adaptive density estimation in deconvolution problems with unknown error distribution , 2013 .

[12]  J. Torrea,et al.  Sobolev spaces associated to the harmonic oscillator , 2006, math/0608684.

[13]  C. Butucea.  Deconvolution of supersmooth densities with smooth noise , 2004 .

[14]  B. Vidakovic,et al.  Adaptive wavelet estimator for nonparametric density deconvolution , 1999 .

[15]  G. Mabon.  Adaptive Deconvolution on the Non‐negative Real Line , 2017 .

[16]  Richard Askey,et al.  Mean convergence of expansions in Laguerre and Hermite series , 1965 .

[17]  Jianqing Fan.  On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems , 1991 .

[18]  A. Meister.  Deconvolution Problems in Nonparametric Statistics , 2009 .