On Bayesian testimation and its application to wavelet thresholding

• 成果类型: Article
• 署名作者: Abramovich, Felix; Grinshtein, Vadim; Petsa, Athanasia; Sapatinas, Theofanis
• 署名单位: Tel Aviv University; Open University Israel; University of Cyprus
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asp080
• 发表日期: 2010
• 页码: 181198
• 关键词: Nonparametric regression; shrinkage; optimality; RISK

摘要

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 l(p)-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 l(p)-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.