Wavelet Thresholding in Fixed Design Regression for Gaussian Random Fields

In this paper, we consider block thresholding wavelet estimators of spatial regression functions on stationary Gaussian random fields observed over a rectangular domain indexed with Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Z}}}^2$$\end{document}, whose covariance function is assumed to satisfy some weak condition. We investigate their asymptotic rates of convergence under the mean integrated squared error when spatial regression functions belong to a large range of Besov function classes Bp,qs(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{s}_{p,q}({{\mathbb {R}}}^2)$$\end{document}. To do this, we derived a result showing the discrepancy between empirical wavelet coefficients and true wavelet coefficients is within certain small rate across above Besov function classes. Based on that, we are able to determine the rates of convergence of our estimators and the supremum norm error over above function classes. The obtained rates of convergence correspond to those established in the standard univariate nonparametric regression with short-range dependence. Therefore, those rates could be considered as sharp as possible. A mild simulation study is carried out to examine the finite sample performance of the proposed estimates.