Opuscula Math. 40, no. 3 (2020), 361-373
https://doi.org/10.7494/OpMath.2020.40.3.361
Opuscula Mathematica
A note on confidence intervals for deblurred images
Abstract. We consider pointwise asymptotic confidence intervals for images that are blurred and observed in additive white noise. This amounts to solving a stochastic inverse problem with a convolution operator. Under suitably modified assumptions, we fill some apparent gaps in the proofs published in [N. Bissantz, M. Birke, Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators, J. Multivariate Anal. 100 (2009), 2364-2375]. In particular, this leads to modified bootstrap confidence intervals with much better finite-sample behaviour than the original ones, the validity of which is, in our opinion, questionable. Some simulation results that support our claims and illustrate the behaviour of the confidence intervals are also presented.
Keywords: inverse problems, confidence intervals, convolution, deblurring.
Mathematics Subject Classification: 62G08, 62G15, 62G20.
- P.J. Bickel, M. Rosenblatt, On some global measures of the deviations of density function estimates, Ann. Statist. 1 (1973), 1071-1095.
- M. Birke, N. Bissantz, H. Holzmann, Confidence bands for inverse regression models, Inverse Problems 26 (2010), Article 115020.
- N. Bissantz, M. Birke, Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators, J. Multivariate Anal. 100 (2009), 2364-2375.
- N. Bissantz, H. Holzmann, Statistical inference for inverse problems, Inverse Problems 24 (2008), Article 034009.
- N. Bissantz, H. Holzmann, K. Proksch, Confidence regions for images observed under the Radon transform, J. Multivariate Anal. 128 (2014), 86-107.
- N. Bissantz, L. Dümbgen, H. Holzman, A. Munk, Non-parametric confidence bands in deconvolution density estimation, J. Roy. Statist. Soc. B 69 (2007), 483-506.
- N. Bissantz, T. Hohage, A. Munk, F.H. Ruymgaart, Convergence rates of general regularization methods for statistical inverse problems and applications, SIAM. J. Numer. Anal. 45 (2007), 2610-2636.
- G. Blanchard, M. Hoffman, M. Reiss, Optimal adaptation for early stopping in statistical inverse problems, SIAM/ASA J. Uncertain. Quantif. 6 (2018), 1043-1075.
- L. Cavalier, Nonparametric statistical inverse problems, Inverse Problems 24 (2008), Article 034004.
- B. Ćmiel, Z. Szkutnik, J. Wojdyła, Asymptotic confidence bands in the Spektor-Lord-Willis problem via kernel estimation of intensity derivative, Electron. J. Stat. 12 (2018), 194-223.
- A. Delaigle, P. Hall, F. Jamshidi, Confidence bands in non-parametric error-in-variables regression, J. Roy. Statist. Soc. B 77 (2015), 149-169.
- A.K. Dey, F.H. Ruymgaart, Direct density estimation as an ill-posed inverse estimation problem, Stat. Neerl. 53 (1999), 309-326.
- H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Mathematics and Its Applications, vol. 375, Kluwer Academic, Dordrecht, 1996.
- E. Giné, R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, New York, 2016.
- K. Lounici, R. Nickl, Global uniform risk bounds for wavelet deconvolution estimator, Ann. Statist. 39 (2011), 201-231.
- B.A. Mair, F.H. Ruymgaart, Statistical inverse estimation in Hilbert scales, SIAM J. Appl. Math. 56 (1996), 1424-1444.
- D.T.P. Nguyen, D. Nuyens, Multivariate integration over \(\mathbb{R}^s\) with exponential rate of convergence, J. Comput. Appl. Math. 315 (2016), 327-342.
- K. Proksch, N. Bissantz, H. Dette, Confidence bands for multivariate and time dependent inverse regression models, Bernoulli 21 (2015), 144-175.
- J. Wojdyła, Z. Szkutnik, Nonparametric confidence bands in Wicksell's problem, Statist. Sinica 28 (2018), 93-113.
- Michał Biel
- Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland
- Zbigniew Szkutnik (corresponding author)
https://orcid.org/0000-0002-4607-6268- Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland
- Communicated by Mirosław Pawlak.
- Received: 2019-04-01.
- Revised: 2019-11-12.
- Accepted: 2020-02-04.
- Published online: 2020-04-04.
