Article

Log-Density Deconvolution by Wavelet Thresholding

Institut d'�conomie Industrielle (IDEI), Toulouse, IDEI Working Papers 01/2009;
Source: RePEc

ABSTRACT This paper proposes a new wavelet-based method for deconvolving a density. The estimator combines the ideas of nonlinear wavelet thresholding with periodised Meyer wavelets and estimation by information projection. It is guaranteed to be in the class of density functions, in particular it is positive everywhere by construction. The asymptotic optimality of the estimator is established in terms of rate of convergence of the Kullback-Leibler discrepancy over Besov classes. Finite sample properties is investigated in detail, and show the excellent empirical performance of the estimator, compared with other recently introduced estimators.

0 Followers
 · 
83 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider the problem of estimating a density $f_X$ using a sample $Y_1,...,Y_n$ from $f_Y=f_X\star f_{\epsilon}$, where $f_{\epsilon}$ is an unknown density. We assume that an additional sample $\epsilon_1,...,\epsilon_m$ from $f_{\epsilon}$ is observed. Estimators of $f_X$ and its derivatives are constructed by using nonparametric estimators of $f_Y$ and $f_{\epsilon}$ and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density $f_{\epsilon}$, where it is assumed that $f_X$ satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density $f_X$ belongs to a Sobolev space $H_{\mathbh p}$ and $f_{\epsilon}$ is ordinary smooth or supersmooth. Comment: Published in at http://dx.doi.org/10.1214/08-AOS652 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
    The Annals of Statistics 05/2007; 37(5). DOI:10.1214/08-AOS652 · 2.44 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: In the article, the problem of the estimation of some unknown density function f over interval [0, 1] is addressed. Given an i.i.d. sample drawn from f and a list of parametric models we propose model selection procedure based on the thresholding of the estimators of parameters. Consistency of the method and consistency of post-model-selection density estimator is derived in the cases when either f(x) or its logarithm has a representation as (possibly infinite) linear combination of orthonormal functions in L 2([0, 1]). The number of competing models is allowed to grow with the number of observations to infinity. We also present the results of simulation study when Legendre polynomials are considered as functions (b j ).
    Communication in Statistics- Theory and Methods 09/2011; 40(17):3082-3098. DOI:10.1080/03610926.2010.491593 · 0.28 Impact Factor

Preview

Download
3 Downloads
Available from