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Qing Liu Accepted Dissertation 3-22-19 Sp 19.pdf (563.73 KB)
ETD Abstract Container
Abstract Header
On Anisotropic Functional Fourier Deconvolution Problem with Unknown Kernel
Author Info
Liu, Qing
ORCID® Identifier
http://orcid.org/0000-0001-6020-4097
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1553711028518802
Abstract Details
Year and Degree
2019, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
Abstract
We consider the estimation of a periodic bivariate function ƒ(⋅,⋅) based on observations from a noisy convolution model, when the kernel (blurring) function 𝘨(⋅,⋅) is unknown. However, we are able to observe 𝘨
δ
(⋅,⋅), a noisy version of 𝘨, at the same time, which assures the estimation. We perform the deconvolution algorithm in Fourier domain. A preliminary thresholding procedure is applied to the Fourier coefficients of observations 𝘨
δ
to ensure a stable inversion. We construct a hard-thresholding wavelet estimator of ƒ using band-limited wavelet bases together with compactly supported wavelet bases so that a fast estimation algorithm exists. To evaluate the performance of our estimator, we derive the lower bounds for the mean integrated squared error (𝐿
2
-risk) assuming that ƒ belongs to the Besov space of mixed smoothness and the kernel 𝘨 possesses certain smoothness properties. We show that the proposed wavelet estimator is adaptive and asymptotically quasi-optimal within a logarithmic factor (in the minimax sense) in a wide range of Besov balls. Furthermore, we investigate the discrete case of our deconvolution model, as it is the common case in real life. We carry out a limited simulation study and show that our estimator performs well in a finite sample setting. Finally, we extend the minimax results to the more general 𝐿
𝑝
-risk (1 ≤ 𝑝 < ∞), and show that our estimator is asymptotically quasi-optimal within a logarithmic factor in this case as well.
Committee
Rida Benhaddou (Advisor)
Wei Lin (Committee Member)
Chang Liu (Committee Member)
Vladimir Vinogradov (Committee Member)
Pages
101 p.
Subject Headings
Mathematics
;
Statistics
Keywords
Anisotropic functional data
;
blind deconvolution
;
Besov space
;
minimax convergence rates
;
hard-thresholding
;
adaptivity
;
wavelets
Recommended Citations
Refworks
EndNote
RIS
Mendeley
Citations
Liu, Q. (2019).
On Anisotropic Functional Fourier Deconvolution Problem with Unknown Kernel
[Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1553711028518802
APA Style (7th edition)
Liu, Qing.
On Anisotropic Functional Fourier Deconvolution Problem with Unknown Kernel.
2019. Ohio University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1553711028518802.
MLA Style (8th edition)
Liu, Qing. "On Anisotropic Functional Fourier Deconvolution Problem with Unknown Kernel." Doctoral dissertation, Ohio University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1553711028518802
Chicago Manual of Style (17th edition)
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Document number:
ohiou1553711028518802
Download Count:
321
Copyright Info
© 2019, all rights reserved.
This open access ETD is published by Ohio University and OhioLINK.