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Sharp adaptive estimation by a blockwise method

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  • T. Tony Cai
  • Mark Low
  • Linda Zhao

Abstract

We consider a blockwise James–Stein estimator for nonparametric function estimation in suitable wavelet or Fourier bases. The estimator can be readily explained and implemented. We show that the estimator is asymptotically sharp adaptive in minimax risk over any Sobolev ball containing the true function. Further, for a moderately broad range of bounded sets in the Besov space our estimator is asymptotically nearly sharp adaptive in the sense that it comes within the Donoho–Liu constant, 1.24, of being exactly sharp adaptive. Other parameter spaces are also considered. The paper concludes with a Monte-Carlo study comparing the performance of our estimator with that of three other popular wavelet estimators. Our procedure generally (but not always) outperforms two of these and is overall comparable, or perhaps slightly superior, with the third.

Suggested Citation

  • T. Tony Cai & Mark Low & Linda Zhao, 2009. "Sharp adaptive estimation by a blockwise method," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(7), pages 839-850.
    • Handle: RePEc:taf:gnstxx:v:21:y:2009:i:7:p:839-850
    DOI: 10.1080/10485250903026320
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    References listed on IDEAS

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    1. Goldenshluger, Alexander & Tsybakov, Alexandre, 2003. "Optimal prediction for linear regression with infinitely many parameters," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 40-60, January.
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    Cited by:

    1. Jiang, Wenhua & Zhang, Cun-Hui, 2013. "A nonparametric empirical Bayes approach to adaptive minimax estimation," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 82-95.
    2. Tuvaandorj, Purevdorj, 2020. "Regression discontinuity designs, white noise models, and minimax," Journal of Econometrics, Elsevier, vol. 218(2), pages 587-608.

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