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A MOLLIFIER APPROACH TO THE DECONVOLUTION OF PROBABILITY DENSITIES

Published online by Cambridge University Press:  28 October 2022

Thorsten Hohage ,
Pierre Maréchal ,
Léopold Simar  and
Anne Vanhems
Thorsten Hohage
Affiliation:
Institute for Numerical and Applied Mathematics, University of Göttingen
Pierre Maréchal
Affiliation:
Mathematical Institute of Toulouse, Paul Sabatier University
Léopold Simar
Affiliation:
Institut de Statistique, Biostatistique et sciences actuarielles, Université Catholique de Louvain-la-Neuve
Anne Vanhems*
Affiliation:
TBS Business School
*
Address correspondence to Anne Vanhems, TBS Business School, Toulouse, France; e-mail: [email protected].
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Abstract

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We use mollification to regularize the problem of deconvolution of random variables. This regularization method offers a unifying and generalizing framework in order to compare the benefits of various filter-type techniques like deconvolution kernels, Tikhonov, or spectral cutoff methods. In particular, the mollifier approach allows to relax some restrictive assumptions required for the deconvolution kernels, and has better stabilizing properties compared with spectral cutoff or Tikhonov. We show that this approach achieves optimal rates of convergence for both finitely and infinitely smoothing convolution operators under Besov and Sobolev smoothness assumptions on the unknown probability density. The qualification can be arbitrarily high depending on the choice of the mollifier function. We propose an adaptive choice of the regularization parameter using the Lepskiĭ method, and we provide simulations to compare the finite sample properties of our estimator with respect to the well-known regularization methods.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 2 , April 2024 , pp. 320 - 359
Copyright
© The Author(s), 2022. Published by Cambridge University Press

References

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