Density deconvolution under general assumptions on the distribution of measurement errors

Denis Belomestny, Alexander Goldenshluger

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we study the problem of density deconvolution under general assumptions on the measurement error distribution. Typically, deconvolution estimators are constructed using Fourier transform techniques, and it is assumed that the characteristic function of the measurement errors does not have zeros on the real line. This assumption is rather strong and is not fulfilled in many cases of interest. In this paper, we develop a methodology for constructing optimal density deconvolution estimators in the general setting that covers vanishing and nonvanishing characteristic functions of the measurement errors. We derive upper bounds on the risk of the proposed estimators and provide sufficient conditions under which zeros of the corresponding characteristic function have no effect on estimation accuracy. Moreover, we show that the derived conditions are also necessary in some specific problem instances.

Original languageEnglish
Pages (from-to)615-649
Number of pages35
JournalAnnals of Statistics
Volume49
Issue number2
DOIs
StatePublished - 2021

Bibliographical note

Funding Information:
which is equivalent to 2r ≤ 2m. Therefore if p < 2r − 1 ≤ 2m − 1 then the standard rate of convergence is not attained. This completes the proof. □ Acknowledgments. The authors are grateful to Taeho Kim for careful reading and useful remarks. This article was prepared within the framework of the HSE University Basic Research Program. Alexander Goldenshluger is supported by the ISF Research Grant no. 361/15. Denis Belomestny acknowledges the financial support from Deutsche Forschungsgemeinschaft (DFG) through the SFB 823 “Statistical modeling of nonlinear dynamic processes.”

Publisher Copyright:
© Institute of Mathematical Statistics, 2021

Keywords

  • Characteristic function
  • Density deconvolution
  • Density estimation
  • Laplace transform
  • Lower bounds
  • Minimax risk

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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