Abstract
The purpose of this paper is the pointwise estimation of a multivariate probability density function with weighted distribution using wavelet methods. New theoretical contributions are provided; Point-wise convergence rates of wavelet estimators are established in the local Hölder space. First, a lower bound is provided for all the possible estimators. Specially, a linear wavelet estimator is defined and turned out to be the optimal one in the considered setting. Subsequently, regarding the adaptive estimation problem, a nonlinear estimator is proposed as usual and discussed. Finally, a new data driven wavelet estimator is introduced and shown to be completely adaptive and almost optimal.
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Acknowledgements
The authors are grateful to the anonymous referee for valuable comments that helped to improve the quality of the article.
Funding
This work is supported by the National Natural Science Foundation of China (Nos. 12101014, 12001133 and 12001132) and the Natural Science Basic Research plan in Shaanxi Province of China (No.2020JQ-892).
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Appendix
Appendix
Here we present the Bernstein inequality used in this study.
Bernstein’s inequality [11]. Let \(X_{1},\ldots ,X_n\) be a sequence of independent random variables such that \(\textbf{E}(X_{i})=0,\;\textbf{E}(X^2_{i})=\sigma ^2\) and \(|X_i|\le M'<\infty \) (\(i=1,\ldots ,n\)). Then for each \(\vartheta >0,\) we have
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Chen, L., Chesneau, C., Kou, J. et al. Wavelet Optimal Estimations for a Multivariate Probability Density Function Under Weighted Distribution. Results Math 78, 66 (2023). https://doi.org/10.1007/s00025-023-01846-1
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DOI: https://doi.org/10.1007/s00025-023-01846-1
Keywords
- Point-wise function estimation
- wavelet estimator
- multivariate probability density function
- weighted distribution
- data driven
- hölder space