Adaptive and optimal pointwise deconvolution density estimations by wavelets

Abstract

This paper considers multivariate deconvolution density estimations under the local Hölder condition by wavelet methods. A pointwise lower bound of the deconvolution model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate. The nonlinear wavelet estimator is introduced for adaptivity, which attains a nearly optimal rate (optimal up to a logarithmic factor). Because the nonlinear wavelet estimator depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data-driven version without any assumption on the estimated functions.

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Acknowledgements

The authors would like to thank the referees for their valuable suggestions, which greatly improve the readability of the article. The authors also thank Prof. Youming Liu (Beijing University of Technology, China) and Prof. Huifang Jia (Shanxi University, China) for their important comments and suggestions.

Funding

This work is supported by the National Natural Science Foundation of China (No. 11901019), and the Science and Technology Program of Beijing Municipal Commission of Education (No. KM202010005025).

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Correspondence to Xiaochen Zeng.

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Communicated by: Yuesheng Xu

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Wu, C., Zeng, X. & Mi, N. Adaptive and optimal pointwise deconvolution density estimations by wavelets. Adv Comput Math 47, 14 (2021). https://doi.org/10.1007/s10444-021-09844-z

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Keywords

  • Wavelets
  • Thresholding
  • Data driven
  • Deconvolution
  • Density estimation

Mathematics subject classification (2010)

  • 42C40
  • 62G07
  • 62G20