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On the estimation of the function and its derivatives in nonparametric regression: a Bayesian testimation approach. (English) Zbl 1418.62169

Summary: We consider the problem of estimating the unknown response function and its derivatives in the standard nonparametric regression model. Recently in [Biometrika 97, No. 1, 181–198 (2010; Zbl 1183.62042)], F. Abramovich et al. applied a Bayesian testimation procedure in a wavelet context and proved asymptotical minimaxity of the resulting adaptive level-wise maximum a posteriori wavelet testimator of the unknown response function and its derivatives in the Gaussian white noise model. Using the boundary-modified coiflets of I. M. Johnstone and B. W. Silverman [J. Appl. Probab. 41A, 81–98 (2004; Zbl 1049.62041)], we show that discretization of the data does not affect the order of magnitude of the accuracy of a discrete version of the suggested level-wise maximum a posteriori wavelet testimator, obtaining thus its adaptivity and asymptotical minimaxity in the standard nonparametric regression model that is usually considered in practical applications. Simulated examples are used to illustrate the performance of the developed wavelet testimation procedure and compared with three recently proposed empirical Bayes wavelet estimators and a block thresholding wavelet estimator.

MSC:

62G08 Nonparametric regression and quantile regression
62G07 Density estimation
62C10 Bayesian problems; characterization of Bayes procedures

Software:

reccv; EBayesThresh
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Full Text: DOI

References:

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