Wavelet Pointwise Estimations Of The Derivative Functions Under Multichannel Gaussian White Noise Model

The deconvolution model with Gaussian white noise has important theoreti-cal and application significances in statistics and many other practical problems.The classical kernel method restricts its application due to the complexity of band-width selection.Wavelet method has been successfully applied to deconvolution estimation because of its outstanding ability of time-frequency analysis.Inspired by the work of F.Navarro etc.(F.Navarro,C.Chesneau,J.Fadili and T.Sassi.Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model.Electronic Journal of Statistics,2013,7:428-453.),this paper discusses the wavelet estimations of the unknown derivative functions of a class of Gaussian white noise model.Specifically,we observe the heteroscedastic stochastic process Yv(t),satisfyingdYv(t)=(f(?)gv)(t)dt + ?dWv(t),for any v ? 1,...,n} and t ?[0,1],where ?>0 is the noise level.gv(t)is a known blurring function.Wv(t)is a standard Brownian motion.The aim is to estimate the d-th derivative of f(t)from the known information of the stochastic process Yv(t).We propose linear and non-linear wavelet estimators,and investigate the pointwise convergences over Holder smoothness spaces.The results show that the linear and non-linear estimators have the same convergence order up to a logarithmic factor.Moreover,the non-linear wavelet estimators based on hard thresholding is an adaptive estimator.In order to show the effectiveness of our theoretical results,some simulation studies are given at last.The experiments show that the wavelet estimators can approximate the tested functions.Moreover,the estimation performances become worse as the order of the derivative increases.This phenomenon is consistent with the former theoretical results.