The Annals of Statistics

On deconvolution of distribution functions

I. Dattner, A. Goldenshluger, and A. Juditsky

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Abstract

The subject of this paper is the problem of nonparametric estimation of a continuous distribution function from observations with measurement errors. We study minimax complexity of this problem when unknown distribution has a density belonging to the Sobolev class, and the error density is ordinary smooth. We develop rate optimal estimators based on direct inversion of empirical characteristic function. We also derive minimax affine estimators of the distribution function which are given by an explicit convex optimization problem. Adaptive versions of these estimators are proposed, and some numerical results demonstrating good practical behavior of the developed procedures are presented.

Article information

Source
Ann. Statist. Volume 39, Number 5 (2011), 2477-2501.

Dates
First available in Project Euclid: 30 November 2011

Permanent link to this document
http://projecteuclid.org/euclid.aos/1322663465

Digital Object Identifier
doi:10.1214/11-AOS907

Mathematical Reviews number (MathSciNet)
MR2906875

Zentralblatt MATH identifier
06008043

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties

Keywords
Adaptive estimator deconvolution minimax risk rates of convergence distribution function

Citation

Dattner, I.; Goldenshluger, A.; Juditsky, A. On deconvolution of distribution functions. Ann. Statist. 39 (2011), no. 5, 2477--2501. doi:10.1214/11-AOS907. http://projecteuclid.org/euclid.aos/1322663465.


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Supplemental materials

  • Supplementary material: Supplement to “On deconvolution of distribution functions”. In the supplementary paper [8] we prove results stated here and provide additional details for the proofs appearing in Section 5. In particular, [8] is partitioned in two Appendices, A and B. Appendix A contains proofs for Section 2: full technical details for Theorems 2.1, 2.3 and 2.4 are presented, and the proof of Theorem 2.2 is given. In Appendix B we prove Theorem 3.1 from Section 3.