Abstract
In this paper we consider the convolutionmodel Z = X + Y withX of unknown density f, independent of Y, when both random variables are nonnegative. Our goal is to estimate the unknown density f of X from n independent identically distributed observations of Z, when the law of the additive process Y is unknown. When the density of Y is known, a solution to the problem has been proposed in [17]. To make the problem identifiable for unknown density of Y, we assume that we have access to a preliminary sample of the nuisance process Y. The question is to propose a solution to an inverse problem with unknown operator. To that aim, we build a family of projection estimators of f on the Laguerre basis, well-suited for nonnegative random variables. The dimension of the projection space is chosen thanks to a model selection procedure by penalization. At last we prove that the final estimator satisfies an oracle inequality. It can be noted that the study of the mean integrated square risk is based on Bernstein’s type concentration inequalities developed for random matrices in [23].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (Dover, New York, 1964).
D. Belomestny, F. Comte, and V. Genon-Catalot, “Nonparametric Laguerre Estimation in the Multiplicative CensoringModel”, Electron. J. Statist. 10 (2), 3114–3152 (2016).
B. Bongioanni and J. L. Torrea, “What is a Sobolev Space for the Laguerre Function Systems?”, Studia Math. 192 (2):147–172 (2009).
R. Y. Chen, A. Gittens, and J. A. Tropp, “The Masked Sample Covariance Estimator: An Analysis Using Matrix Concentration Inequalities”, Inf. Inference 1 (1), 2–20 (2012).
F. Comte, C. A. Cuenod, M. Pensky, and Y. Rozenholc, “Laplace Deconvolution and Its Application to Dynamic Contrast Enhanced Imaging”, J. Roy. Statist. Soc., Ser. B (2015).
F. Comte and V. Genon-Catalot, Adaptive Laguerre Density Estimation for Mixed PoissonModels. Electron. J. Statist. 9, 1112–1148 (2015).
F. Comte and V. Genon-Catalot, Laguerre and Hermite bases for inverse problems, Preprint hal-01449799, V2 (2017). https://hal.archives-ouvertes.fr/hal-01449799.
F. Comte and C. Lacour, “Data-Driven Density Estimation in the Presence of Additive Noise with Unknown Distribution”, J. Roy. Statist. Soc., Ser. B 73, 601–627 (2011).
I. Dattner, M. Reiβ, and M. Trabs, “Adaptive Quantile Estimation in Deconvolution with Unknown Error Distribution”, Bernoulli 22 (2), 143–192 (2016).
P. Groeneboom and G. Jongbloed, “Density Estimation in the UniformDeconvolution Model”, Statist. Neerl. 57 (1), 136–157 (2003).
P. Groeneboom and J. A. Wellner, Information Bounds and Nonparametric Maximum Likelihood Estimation in DMV Seminar (Birkhäuser, Basel, 1992), vol.19.
R. Horn and C. Johnson, Matrix Analysis (Cambridge University Press, 1990).
J. Johannes, “Deconvolution with Unknown Error Distribution”, Ann. Statist. 37 (5a), 2301–2323 (2009).
J. Johannes and M. Schwarz, “Adaptive Circular Deconvolution by Model Selection under Unknown Error Distribution”, Bernoulli 19 (5A), 1576–1611 (2013).
J. Kappus and G. Mabon, “Adaptive Density Estimation in Deconvolution Problems with Unknown Error Distribution”, Electron. J. Statist. 8 (2), 2879–2904 (2014).
G. Mabon, “Adaptive Deconvolution of Linear Functionals on the Nonnegative Real Line”, J. Statist. Plann. Inference 178, 1–23 (2016).
G. Mabon, “Adaptive Deconvolution on the Nonnegative Real Line”, Scandinavian J. Statist. 44, 707–740 (2017).
P. Massart, Concentration Inequalities and Model Selection, in Lecture Notes in Mathematics, Vol. 1896: Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003 (Springer, Berlin, 2003).
A. Meister, Deconvolution Problems in Nonparametric Statistics, in Lecture Notes in Statistics (Springer, Berlin, 2009), Vol.193.
M. H. Neumann, “On the Effect of Estimating the Error Density in Nonparametric Deconvolution”, J. Nonparametric Statist. 7 (4), 307–330 (1997).
G.W. Stewart and J.-G. Sun, Matrix Perturbation Theory (Academic Press, Boston etc., 1990).
J. A. Tropp, “User-Friendly Tail Bounds for Sums of Random Matrices”, Found. Comput. Math. 12 (4), 389–434 (2012).
J. A. Tropp, “An Introduction to Matrix Concentration Inequalities”, Found. Trends Mach. Learn. 8 (1–2), 1–230 (2015).
B. van Es, “Combining Kernel Estimators in the Uniform Deconvolution Problem”, Statist. Neerl. 65 (3), 275–296 (2011).
T. Vareschi, “Noisy Laplace Deconvolutionwith Error in the Operator”, J. Statist. Plann. Inference 157–158, 16–35 (2015).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Comte, F., Mabon, G. Laguerre deconvolution with unknown matrix operator. Math. Meth. Stat. 26, 237–266 (2017). https://doi.org/10.3103/S1066530717040019
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530717040019
Keywords
- convolution model
- linear inverse problem
- nonnegative random variables
- Laguerre basis
- nonparametric density estimation
- random matrix
- oracle inequalities
- adaptive estimation