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Volume 40, Issue 1
Adaptive and Optimal Point-Wise Estimations for Densities in GARCH-Type Model by Wavelets

Cong Wu, Jinru Wang & Xiaochen Zeng

DOI: 10.4208/jcm.2007-m2020-0109

J. Comp. Math., 40 (2022), pp. 108-126.

Published online: 2021-11

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  • Abstract

This paper considers adaptive point-wise estimations of density functions in GARCH-type model under the local Hölder condition by wavelet methods. A point-wise lower bound estimation of that model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate, which means that the convergence rate coincides with the lower bound. The non-linear wavelet estimator is introduced for adaptivity, although it is nearly-optimal. However, the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data driven version without any assumptions on the estimated functions.

  • Keywords

Wavelets, Point-wise risk, Thresholding, Data-driven, GARCH-type model.

  • AMS Subject Headings

42C40, 62G07, 62G20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wuc@hbut.edu.cn (Cong Wu)

wangjinru@bjut.edu.cn (Jinru Wang)

zengxiaochen@bjut.edu.cn (Xiaochen Zeng)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-108, author = {Wu , CongWang , Jinru and Zeng , Xiaochen}, title = {Adaptive and Optimal Point-Wise Estimations for Densities in GARCH-Type Model by Wavelets}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {40}, number = {1}, pages = {108--126}, abstract = {

This paper considers adaptive point-wise estimations of density functions in GARCH-type model under the local Hölder condition by wavelet methods. A point-wise lower bound estimation of that model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate, which means that the convergence rate coincides with the lower bound. The non-linear wavelet estimator is introduced for adaptivity, although it is nearly-optimal. However, the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data driven version without any assumptions on the estimated functions.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2007-m2020-0109}, url = {http://global-sci.org/intro/article_detail/jcm/19972.html} }
TY - JOUR T1 - Adaptive and Optimal Point-Wise Estimations for Densities in GARCH-Type Model by Wavelets AU - Wu , Cong AU - Wang , Jinru AU - Zeng , Xiaochen JO - Journal of Computational Mathematics VL - 1 SP - 108 EP - 126 PY - 2021 DA - 2021/11 SN - 40 DO - http://doi.org/10.4208/jcm.2007-m2020-0109 UR - https://global-sci.org/intro/article_detail/jcm/19972.html KW - Wavelets, Point-wise risk, Thresholding, Data-driven, GARCH-type model. AB -

This paper considers adaptive point-wise estimations of density functions in GARCH-type model under the local Hölder condition by wavelet methods. A point-wise lower bound estimation of that model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate, which means that the convergence rate coincides with the lower bound. The non-linear wavelet estimator is introduced for adaptivity, although it is nearly-optimal. However, the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data driven version without any assumptions on the estimated functions.

Cong Wu, Jinru Wang & Xiaochen Zeng. (2021). Adaptive and Optimal Point-Wise Estimations for Densities in GARCH-Type Model by Wavelets. Journal of Computational Mathematics. 40 (1). 108-126. doi:10.4208/jcm.2007-m2020-0109
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