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Nonparametric regression on random fields with random design using wavelet method

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Abstract

We consider non-linear wavelet-based estimators of spatial regression functions with (known) random design on strictly stationary random fields, which are indexed by the integer lattice points in the \(N\)-dimensional Euclidean space and are assumed to satisfy some mixing conditions. We investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients and show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, wavelet estimators still achieve nearly optimal convergence rates for random fields and provide explicitly the extraordinary local adaptability.

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Acknowledgments

The author is very 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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  1. Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824, USA

    Linyuan Li

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  1. Linyuan Li
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Correspondence to Linyuan Li.

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Li, L. Nonparametric regression on random fields with random design using wavelet method. Stat Inference Stoch Process 19, 51–69 (2016). https://doi.org/10.1007/s11203-015-9119-8

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  • DOI: https://doi.org/10.1007/s11203-015-9119-8

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