Abstract
In this paper, we establish wavelet estimations on pointwise \(l^{p}(1\le p<\infty )\) risk for multivariate regression functions based on biased data. We firstly introduce a linear wavelet estimator and discuss the convergence rate of this estimator. In order to obtain an adaptive estimator, a nonlinear wavelet estimator is constructed by the hard thresholding method. It should be pointed out that the convergence rate of linear and nonlinear wavelet estimators coincide with the optimal convergence rate of pointwise nonparametric estimation.
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Acknowledgements
The authors would like to thank the anonymous reviewer for the important comments and suggestions. This paper is supported by Guangxi Natural Science Foundation (Nos. 2017GXNSFAA198194 and 2018GXNSFBA281076), Guangxi Science and Technology Project (Nos. Guike AD18281058 and Guike AD18281019), the Guangxi Young Teachers Basic Ability Improvement Project (Nos. 2018KY0212 and 2019KY0218) and Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation.
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Guo, H., Kou, J. Pointwise Wavelet Estimation of Regression Function Based on Biased Data. Results Math 74, 128 (2019). https://doi.org/10.1007/s00025-019-1054-3
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DOI: https://doi.org/10.1007/s00025-019-1054-3