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 non-zero 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 levelwise 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.

To cite this version:Athanasia Petsa, Theofanis Sapatinas. Minimax convergence rates under the -risk in the functional deconvolution model. Statistics and Probability Letters, Elsevier, 2009, 79 (13) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A C C E P T E D M A N U S C R I P T

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We consider the problem of estimating the integral of the square of a probability density function f on the basis of a random sample from a weighted distribution. Specifically, using model selection via a penalized criterion, an adaptive estimator for f 2 based on weighted data is proposed for probability density functions which are uniformly bounded and belong to certain Besov bodies. We show that the proposed estimator attains the minimax rate of convergence that is optimal in the case of direct data. Additionally, we obtain the information bound for the problem of estimating f 2 when weighted data are available and compare it with the information bound for the case of direct data. A small simulation study is conducted to illustrate the usefulness of the proposed estimator in finite sample situations.

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