Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T18:42:52.410Z Has data issue: false hasContentIssue false
  • English
  • Français

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].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

Alibaud, N., Maréchal, P., & Saesor, Y. (2009) A variational approach to the inversion of truncated Fourier operators. Inverse Problems 25(4), 045002.CrossRefGoogle Scholar
Bauer, F. & Lukas, M.A. (2011) Comparing parameter choice methods for regularization of ill-posed problems. Mathematics and Computers in Simulation 81(9), 1795–1841.CrossRefGoogle Scholar
Bauschke, H.H. & Combettes, P.L. (2011) Convex Analysis and Monotone Operator Theory in Hilbert Spaces . CMS Books in Mathematics, vol. 408. Springer.CrossRefGoogle Scholar
Bonnefond, X. & Maréchal, P. (2009) A variational approach to the inversion of some compact operators. Pacific Journal of Optimization 5(1), 97–110.Google Scholar
Butucea, C. (2004) Deconvolution of supersmooth densities with smooth noise. Canadian Journal of Statistics 32(2), 181–192.CrossRefGoogle Scholar
Butucea, C. & Tsybakov, A.B. (2008) Sharp optimality in density deconvolution with dominating bias. I. Theory of Probability & Its Applications 52(1), 24–39.CrossRefGoogle Scholar
Carrasco, M. & Florens, J.-P. (2011) A spectral method for deconvolving a density. Econometric Theory 27(3), 546–581.CrossRefGoogle Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007) Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. Handbook of Econometrics 6, 5633–5751.CrossRefGoogle Scholar
Carroll, R.J. & Hall, P. (1988) Optimal rates of convergence for deconvolving a density. Journal of the American Statistical Association 83(404), 1184–1186.CrossRefGoogle Scholar
Comte, F. & Lacour, C. (2013) Anisotropic adaptive kernel deconvolution. Annales de l’IHP Probabilités et Statistiques 49, 569–609.Google Scholar
Comte, F., Rozenholc, Y., & Taupin, M.-L. (2006) Penalized contrast estimator for adaptive density deconvolution. Canadian Journal of Statistics 34(3), 431–452.CrossRefGoogle Scholar
Comte, F., Rozenholc, Y., & Taupin, M.-L. (2007) Finite sample penalization in adaptive density deconvolution. Journal of Statistical Computation and Simulation 77(11), 977–1000.CrossRefGoogle Scholar
Delaigle, A. & Hall, P. (2016) Methodology for non-parametric deconvolution when the error distribution is unknown. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 78(1), 231–252.CrossRefGoogle Scholar
Delaigle, A., Hall, P., & Meister, A. (2008) On deconvolution with repeated measurements. Annals of Statistics 36(2), 665–685.CrossRefGoogle Scholar
Delaigle, A. & Meister, A. (2011) Nonparametric function estimation under Fourier-oscillating noise. Statistica Sinica 21, 1065–1092.CrossRefGoogle Scholar
Devroye, L. (1989) Consistent deconvolution in density estimation. Canadian Journal of Statistics 17(2), 235–239.CrossRefGoogle Scholar
Efromovich, S. (1997) Density estimation for the case of supersmooth measurement error. Journal of the American Statistical Association 92(438), 526–535.CrossRefGoogle Scholar
Engl, H.W., Hanke, M., & Neubauer, A. (1996) Regularization of Inverse Problems. Mathematics and Its Application, vol. 375. Springer Science & Business Media.CrossRefGoogle Scholar
Fan, J. (1991a) Global behavior of deconvolution kernel estimates. Statistica Sinica 1, 541–551.Google Scholar
Fan, J. (1991b) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19, 1257–1272.CrossRefGoogle Scholar
Friedrichs, K.O. (1944) The identity of weak and strong extensions of differential operators. Transactions of the American Mathematical Society 55(1), 132–151.CrossRefGoogle Scholar
Giné, E. & Nickl, R. (2021) Mathematical Foundations of Infinite-Dimensional Statistical Models . Cambridge University Press.Google Scholar
Hall, P. & Meister, A. (2007) A ridge-parameter approach to deconvolution. Annals of Statistics 35(4), 1535–1558.CrossRefGoogle Scholar
Hào, D.N. (1994) A mollification method for ill-posed problems. Numerische Mathematik 68, 469–506.Google Scholar
Hohage, T. & Weidling, F. (2017) Characterizations of variational source conditions, converse results, and maxisets of spectral regularization methods. SIAM Journal on Numerical Analysis 55(2), 598–620.CrossRefGoogle Scholar
Johannes, J. (2009) Deconvolution with unknown error distribution. Annals of Statistics 37(5A), 2301–2323.CrossRefGoogle Scholar
Kerkyacharian, G. & Picard, D. (1993) Density estimation by kernel and wavelets methods: Optimality of Besov spaces. Statistics & Probability Letters 18(4), 327–336.CrossRefGoogle Scholar
Lannes, A., Roques, S., & Casanove, M.-J. (1987) Stabilized reconstruction in signal and image processing: I. Partial deconvolution and spectral extrapolation with limited field. Journal of Modern Optics 34(2), 161–226.CrossRefGoogle Scholar
Lepskiĭ, O. (1991) On a problem of adaptive estimation in Gaussian white noise. Theory of Probability & Its Applications 35(3), 454–466.CrossRefGoogle Scholar
Lepskiĭ, O. (1992) Asymptotically minimax adaptive estimation. I: Upper bounds. Optimally adaptive estimates. Theory of Probability & Its Applications 36(4), 682–697.CrossRefGoogle Scholar
Lepskiĭ, O. (1993) Asymptotically minimax adaptive estimation. II. Schemes without optimal adaptation: Adaptive estimators. Theory of Probability & Its Applications 37(3), 433–448.CrossRefGoogle Scholar
Lepskiĭ, O. & Willer, T. (2017) Lower bounds in the convolution structure density model. Bernoulli 23(2), 884–926.CrossRefGoogle Scholar
Lepskiĭ, O. & Willer, T. (2019) Oracle inequalities and adaptive estimation in the convolution structure density model. Annals of Statistics 47(1), 233–287.CrossRefGoogle Scholar
Louis, A.K. & Maass, P. (1990) A mollifier method for linear operator equations of the first kind. Inverse Problems 6(3), 427.CrossRefGoogle Scholar
Mair, B.A. & Ruymgaart, F.H. (1996) Statistical inverse estimation in Hilbert scales. SIAM Journal on Applied Mathematics 56(5), 1424–1444.CrossRefGoogle Scholar
Maréchal, P., Togane, D., & Celler, A. (2000) A new reconstruction methodology for computerized tomography: FRECT (Fourier regularized computed tomography). IEEE Transactions on Nuclear Science 47(4), 1595–1601.CrossRefGoogle Scholar
Massart, P. (2000) About the constants in Talagrand’s concentration inequalities for empirical processes. The Annals of Probability 28(2), 863–884.CrossRefGoogle Scholar
Mathé, P. (2006) The Lepskii principle revisited. Inverse Problems 22(3), L11.CrossRefGoogle Scholar
Mathé, P. & Pereverzev, S.V. (2003) Geometry of linear ill-posed problems in variable Hilbert scales. Inverse Problems 19(3), 789.CrossRefGoogle Scholar
McCormick, G. & Tapia, R. (1972) The gradient projection method under mild differentiability conditions. SIAM Journal on Control 10(1), 93–98.CrossRefGoogle Scholar
Morozov, V.A. (1984) Methods for Solving Incorrectly Posed Problems . Springer Science & Business Media.CrossRefGoogle Scholar
Murio, D.A. (2011) The Mollification Method and the Numerical Solution of Ill-Posed Problems . Wiley.Google Scholar
Nikol’skii, S.M. (1951) Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables. Trudy Matematicheskogo Instituta imeni VA Steklova 38, 244–278.Google Scholar
Pensky, M. & Vidakovic, B. (1999) Adaptive wavelet estimator for nonparametric density deconvolution. Annals of Statistics 27(6), 2033–2053.CrossRefGoogle Scholar
Rebelles, G. (2016) Structural adaptive deconvolution under ${\lessdot}_p$ -losses. Mathematical Methods of Statistics 25(1), 26–53.CrossRefGoogle Scholar
Rudin, W. (1970) Real and Complex Analysis: Part 2. McGraw-Hill.Google Scholar
Schuster, T. (2007) The Method of Approximate Inverse: Theory and Applications . Lecture Notes in Mathematics, vol. 1906. Springer.CrossRefGoogle Scholar
Stefanski, L.A. & Carroll, R.J. (1990) Deconvolving kernel density estimators. Statistics 21(2), 169–184.CrossRefGoogle Scholar
Tikhonov, A.N. & Arsenin, V.Y. (1977) Methods for Solving Ill-Posed Problems . Wiley.Google Scholar
Triebel, H. (2006) Theory of Function Spaces III . Monographs in Mathematics, vol. 100. Birkhäuser Basel.Google Scholar
Trong, D.D. & Phuong, C.X. (2015) Ridge-parameter regularization to deconvolution problem with unknown error distribution. Vietnam Journal of Mathematics 43(2), 239–256.CrossRefGoogle Scholar
Van Rooij, A.C.M. & Ruymgaart, F.H. (1991) Regularized deconvolution on the circle and the sphere. In G. Roussas (ed.), Nonparametric Functional Estimation and Related Topics , pp 679–690. Springer, Dordrecht.CrossRefGoogle Scholar
Wikipedia Contributors (2020) Mollifier—Wikipedia, the free encyclopedia (Online; accessed 17 April 2020).Google Scholar