Laguerre and Hermite bases for inverse problems

Abstract We present inverse problems of nonparametric statistics which have a smart solution using projection estimators on bases of functions with non compact support, namely, a Laguerre basis or a Hermite basis. The models are Y i = X i U i , Z i = X i + Σ i , where the X i ’s are i.i.d. with unknown density f , the Σ i ’s are i.i.d. with known density f Σ , the U i ’s are i.i.d. with uniform density on [ 0 , 1 ] . The sequences ( X i ) , ( U i ) , ( Σ i ) are independent. We define projection estimators of f in the two cases of indirect observations of ( X 1 , … , X n ) , and we give upper bounds for their L 2 -risks on specific Sobolev–Laguerre or Sobolev–Hermite spaces. Data-driven procedures are described and proved to perform automatically the bias–variance compromise.

[1]  A. Berlinet,et al.  Nonparametric Curve Estimation , 2004 .

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

[3]  C. Chesneau.  Wavelet Estimation of a Density in a GARCH-type Model , 2013 .

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

[5]  T. Vareschi.  Noisy Laplace deconvolution with error in the operator , 2013, 1303.7437.

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

[7]  A. Goldenshluger,et al.  Nonparametric density estimation from observations with multiplicative measurement errors , 2017, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[8]  Martin B. Hansen,et al.  Multiplicative censoring: density estimation by a series expansion approach , 2001 .

[9]  S. Schwartz.  Estimation of Probability Density by an Orthogonal Series , 1967 .

[10]  Zhimin Zhang,et al.  A new efficient method for estimating the Gerber–Shiu function in the classical risk model , 2018 .

[11]  Lianfen Qian,et al.  Nonparametric Curve Estimation: Methods, Theory, and Applications , 1999, Technometrics.

[12]  Yehuda Vardi,et al.  Large Sample Study of Empirical Distributions in a Random-Multiplicative Censoring Model , 1992 .

[13]  Yehuda Vardi,et al.  Multiplicative censoring, renewal processes, deconvolution and decreasing density: Nonparametric estimation , 1989 .

[14]  A. Juditsky,et al.  On minimax density estimation on \mathbb{R}} , 2004 .

[15]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

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

[17]  Adaptive density estimation using the blockwise Stein method , 2006 .

[18]  C. Chesneau,et al.  MULTIPLICATIVE CENSORING: ESTIMATION OF A DENSITY AND ITS DERIVATIVES UNDER THE Lp-RISK , 2013 .

[19]  Jie Shen,et al.  Stable and Efficient Spectral Methods in Unbounded Domains Using Laguerre Functions , 2000, SIAM J. Numer. Anal..

[20]  J. Torrea,et al.  What is a Sobolev space for the Laguerre function systems , 2008 .

[21]  Adaptive Laguerre density estimation for mixed Poisson models , 2015 .

[22]  M. Carone,et al.  Large-sample study of the kernel density estimators under multiplicative censoring , 2012, 1205.6275.

[23]  M. Jirak,et al.  Adaptive function estimation in nonparametric regression with one-sided errors , 2013, 1305.6430.

[24]  H. Zanten,et al.  Nonparametric volatility density estimation , 2001, math/0107135.

[25]  F. Comte,et al.  Nonparametric density and survival function estimation in the multiplicative censoring model , 2016 .

[26]  S. Efromovich.  Lower bound for estimation of Sobolev densities of order less 12 , 2009 .

[27]  P. Massart,et al.  Concentration inequalities and model selection , 2007 .

[28]  C. Chesneau,et al.  Nonparametric estimation of density under bias and multiplicative censoring via wavelet methods , 2012 .

[29]  Marianna Pensky,et al.  Laplace deconvolution on the basis of time domain data and its application to dynamic contrast‐enhanced imaging , 2014, 1405.7107.

[30]  Nonparametric Laguerre estimation in the multiplicative censoring model , 2016 .

[31]  Marie-Luce Taupin,et al.  Penalized contrast estimator for adaptive density deconvolution , 2006, math/0601091.

[32]  G. Jongbloed.  Exponential deconvolution: two asymptotically equivalent estimators , 1998 .

[33]  G. Walter.  Properties of Hermite Series Estimation of Probability Density , 1977 .

[34]  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.

[35]  Geurt Jongbloed,et al.  Isotonic inverse estimators for nonparametric deconvolution , 1998 .