Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 3184–3213 | Cite as

Wavelet Thresholding in Fixed Design Regression for Gaussian Random Fields

  • Linyuan LiEmail author
  • Kewei Lu
  • Yimin Xiao


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 \({{\mathbb {Z}}}^2\), 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 \(B^{s}_{p,q}({{\mathbb {R}}}^2)\). 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.


Spatial regression Random fields Block thresholding Wavelet estimator Convergence rate Besov space 

Mathematics Subject Classification

Primary: 62G08 Secondary: 62M40 62G20 



The authors are grateful to two referees for their careful reading of an earlier version of the manuscript and for their extremely helpful suggestions. This greatly improved the presentation of the paper.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of New HampshireDurhamUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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