TY - JOUR
T1 - On Bayesian testimation and its application to wavelet thresholding
AU - Abramovich, Felix
AU - Grinshtein, Vadim
AU - Petsa, Athanasia
AU - Sapatinas, Theofanis
PY - 2010/3
Y1 - 2010/3
N2 - 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.
AB - 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.
KW - Adaptive estimation
KW - Besov space
KW - Gaussian sequence model
KW - Gaussian white noise model
KW - Lp-ball
KW - Multiple testing
KW - Thresholding
KW - Wavelet estimation
UR - http://www.scopus.com/inward/record.url?scp=77249153180&partnerID=8YFLogxK
U2 - 10.1093/biomet/asp080
DO - 10.1093/biomet/asp080
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AN - SCOPUS:77249153180
SN - 0006-3444
VL - 97
SP - 181
EP - 198
JO - Biometrika
JF - Biometrika
IS - 1
ER -