On Bayesian testimation and its application to wavelet thresholding

Felix Abramovich*, Vadim Grinshtein, Athanasia Petsa, Theofanis Sapatinas

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the problem of estimating the unknown response function in the Gaussian white noise model. We first utilize the recently developed Bayesian maximum a posteriori testimation procedure of Abramovich et al. (2007) for recovering an unknown high-dimensional Gaussian mean vector. The existing results for its upper error bounds over various sparse lp-balls are extended to more general cases. We show that, for a properly chosen prior on the number of nonzero entries of the mean vector, the corresponding adaptive estimator is asymptotically minimax in a wide range of sparse and dense lp-balls. The proposed procedure is then applied in a wavelet context to derive adaptive global and level-wise wavelet estimators of the unknown response function in the Gaussian white noise model. These estimators are then proven to be, respectively, asymptotically near-minimax and minimax in a wide range of Besov balls. These results are also extended to the estimation of derivatives of the response function. Simulated examples are conducted to illustrate the performance of the proposed level-wise wavelet estimator in finite sample situations, and to compare it with several existing counterparts.

Original languageEnglish
Pages (from-to)181-198
Number of pages18
JournalBiometrika
Volume97
Issue number1
DOIs
StatePublished - Mar 2010

Keywords

  • Adaptive estimation
  • Besov space
  • Gaussian sequence model
  • Gaussian white noise model
  • Lp-ball
  • Multiple testing
  • Thresholding
  • Wavelet estimation

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