Abstract
This paper considers multivariate deconvolution density estimations under the local Hölder condition by wavelet methods. A pointwise lower bound of the deconvolution model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate. The nonlinear wavelet estimator is introduced for adaptivity, which attains a nearly optimal rate (optimal up to a logarithmic factor). Because the nonlinear wavelet estimator depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data-driven version without any assumption on the estimated functions.
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References
Butucea, C.: The adaptive rate of convergence in a problem of pointwise density estimation. Stat. Probab. Lett. 47(1), 85–90 (2000)
Butucea, C.: Exact adaptive pointwise estimation on Sobolev classes of densities. ESAIM–Probab. Stat. 5, 1–31 (2001)
Comte, F., Lacour, C.: Anisotropic adaptive kernel deconvolution. Ann. l’Institut. Henri Poincaré Probab. Stat. 49(2), 569–609 (2013)
Delyon, B., Juditsky, A.: On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3(3), 215–228 (1996)
Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D.: Density estimation by wavelet thresholding. Ann. Stat. 24(2), 508–539 (1996)
Fan, J.Q.: On the optimal rate of convergence for nonparametric deconvolution problems. Ann. Stat. 19(3), 1257–1272 (1991)
Fan, J.Q., Koo, J.-Y.: Wavelet deconvolution. IEEE Trans. Inf. Theory 48(3), 734–747 (2002)
Goldenshluger, A., Lepski, O.: On adaptive minimax density estimation on \(\mathbb {R}^{d}\). Probab. Theory Relat. Fields 159(3–4), 479–543 (2014)
Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A.: Wavelets, Approximation and Statistical Applications. Springer, New York (1998)
Jia, H.F., Li, Y.Z.: Weak (quasi-)affine bi-frames for reducing subspaces of \(L^{2}(\mathbb {R}^{d})\). Sci. China– Math. 58(5), 1005–1022 (2015)
Kerkyacharian, G., Picard, D.: Density estimation in Besov spaces. Stat. Probab. Lett. 13(1), 15–24 (1992)
Li, R., Liu, Y.M.: Wavelet optimal estimations for a density with some additive noises. Appl. Comput. Harmon. Anal. 36(3), 416–433 (2014)
Li, Q., Racine, J.S.: Nonparametric Econometrics: Theory and Practice. University Press Princeton, Princeton (2007)
Liu, Y.M., Wu, C.: Point-wise estimation for anisotropic densities. J. Multivar. Anal. 171, 112–125 (2019)
Liu, Y.M., Zeng, X.C.: Strong Lp convergence of wavelet deconvolution density estimators. Anal. Appl. 16 (2), 183–208 (2018)
Liu, Y.M., Zeng, X.C.: Asymptotic normality of wavelet deconvolution density estimators. Appl. Comput. Harmon. Anal. 48(1), 321–342 (2020)
Lounici, K., Nickl, R.: Global uniform risk bounds for wavelet deconvolution estimators. Ann. Stat. 39(1), 201–231 (2011)
Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)
Pensky, M., Vidakovic, B.: Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Stat. 27(6), 2033–2053 (1999)
Pensky, M.: Density deconvolution based on wavelets with bounded supports. Stat. Probab. Lett. 56(3), 261–269 (2002)
Rebelles, G.: Pointwise adaptive estimation of a multivariate density under independence hypothesis. Bernoulli 21(4), 1984–2023 (2015)
Tsybakov, A.B.: Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, New York (2009)
Walter, G.G.: Density estimation in the presence of noise. Stat. Probab. Lett. 41(3), 237–246 (1999)
Walnut, D.F.: An Introduction to Wavelet Analysis. Birkhäuser, Boston (2004)
Zeng, X.C.: A note on wavelet deconvolution density estimation. Int. J. Wavelets, Multiresolution Inf. Process. 15(6), 1750055, 12 pp (2017)
Zeng, X.C., Wang, J.R.: Wavelet density deconvolution estimations with heteroscedastic measurement errors. Stat. Probab. Lett. 134, 79–85 (2018)
Acknowledgements
The authors would like to thank the referees for their valuable suggestions, which greatly improve the readability of the article. The authors also thank Prof. Youming Liu (Beijing University of Technology, China) and Prof. Huifang Jia (Shanxi University, China) for their important comments and suggestions.
Funding
This work is supported by the National Natural Science Foundation of China (No. 11901019), and the Science and Technology Program of Beijing Municipal Commission of Education (No. KM202010005025).
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Communicated by: Yuesheng Xu
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Wu, C., Zeng, X. & Mi, N. Adaptive and optimal pointwise deconvolution density estimations by wavelets. Adv Comput Math 47, 14 (2021). https://doi.org/10.1007/s10444-021-09844-z
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DOI: https://doi.org/10.1007/s10444-021-09844-z