On the estimation of the function and its derivatives in nonparametric regression: A bayesian testimation approach

Abstract

We consider the problem of estimating the unknown response function and its deriva- tives in the standard nonparametric regression model. Recently, Abramovich et al. (2010) applied a Bayesian testimation procedure in a wavelet context and proved asymptotical minimaxity of the resulting adaptive level-wise maximum a posteri- ori wavelet testimator of the unknown response function and its derivatives in the Gaussian white noise model. Using the boundary-modified coiflets of Johnstone and Silverman (2004), we show that dicretization of the data does not a®ect 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 min- imaxity 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 esti- mator. © 2011, Indian Statistical Institute.