On adaptive wavelet estimation of a quadratic functional from a deconvolution problem

摘要

We observe a stochastic process where a convolution product of an unknown function and a known function is corrupted by Gaussian noise. We wish to estimate the squared mathbbL2{mathbb{L}^2} -norm of the unknown function from the observations. To reach this goal, we develop adaptive estimators based on wavelet and thresholding. We prove that they achieve (near) optimal rates of convergence under the mean squared error over a wide range of smoothness classes.