Minimax estimation for the deconvolution model in random design with long-memory dependent errors

Abstract

. Deconvolution model is investigated when the design points t i are random and follow a bounded density h. We propose a wavelet hard-thresholding estimator, and show that it achieves asymptotically near-optimal convergence rates for the L 2-risk when the response function f belongs to Besov space. We consider the problem under two noise structure; (1) i . i . d . Gaussian errors and (2) long-memory Gaussian errors. At a first stage, we assume h is known. For the i . i . d . Gaussian errors case, our convergence rates are similar to those found in the literature under regular design. For the long memory case, our convergence rates are new. However, at each elbow, the convergence rates will depend on the long-memory parameter only if it is small enough, otherwise, the rates will be identical to those we obtained under i . i . d . errors. At a second stage, we assume h is unknown, construct the usual hard-thresholding estimator, and derive the upper bounds for its L 2-risk under the new settings. The rates seem to depend on the smoothness parameter, γ, of h when γ is relatively small.