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Convergence Rates of Wavelet Density Estimators for Strongly Mixing Samples

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Abstract

This paper considers wavelet estimations for a multivariate density function based on strongly mixing data. We first construct a linear wavelet estimator and provide a convergence rate over \(L^{p} (1\le p<\infty )\) risk in Besov space \(B^{s}_{r,q}(\mathbb {R}^{d})\). However, this estimator depends on the smoothness of density function, which means that the estimator is not adaptive. A nonlinear adaptive wavelet estimator is proposed by thresholding method. Moreover, the convergence rate of nonlinear estimator is better than the linear one in the case of \(r\le p\).

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Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful and constructive comments that contributed to improving the original version of this paper. This paper is supported by the National Natural Science Foundation of China (No. 71961004), Guangxi Natural Science Foundation (Nos. 2017GXNSFAA198194, 2018GXNSFBA281076, 2019GXNSFFA245012), Guangxi Science and Technology Project (Nos. Guike AD18281058 and Guike AD18281019), the Guangxi Young Teachers Basic Ability Improvement Project (Nos. 2018KY0212 and 2019KY0218), Innovation Project of Guet Graduate Education (No. 2018YJCX58), and Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation.

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Correspondence to Junke Kou.

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Communicated by Ahmad Parsian.

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Cui, K., Kou, J. Convergence Rates of Wavelet Density Estimators for Strongly Mixing Samples. Bull. Iran. Math. Soc. 47, 701–723 (2021). https://doi.org/10.1007/s41980-020-00408-3

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  • DOI: https://doi.org/10.1007/s41980-020-00408-3

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