Skip to main content

The Empirical Process of Residuals from an Inverse Regression

Abstract

In this paper we investigate an indirect regression model characterized by the Radon transformation. This model is useful for recovery of medical images obtained by computed tomography scans. The indirect regression function is estimated using a series estimator motivated by a spectral cutoff technique. Further, we investigate the empirical process of residuals from this regression, and show that it satisfies a functional central limit theorem.

This is a preview of subscription content, access via your institution.

References

  1. M. G. Akritas and I. van Keilegom, “Nonparametric Estimation of the Residual Distribution”, Scandinavian J. Statist. 28, 549–567 (2001).

    MathSciNet  Article  MATH  Google Scholar 

  2. M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, “Image Deblurring with Poisson Data: From Cells to Galaxies”, Inverse Problems 25(12), 123006, 26 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  3. P. J. Bickel and M. Rosenblatt, “On Some Global Measures of the Deviations of Density Function Estimates”, Ann. Statist. 1, 1071–1095 (1973).

    MathSciNet  Article  MATH  Google Scholar 

  4. N. Bissantz, J. Chown, and H. Dette, “Regularization Parameter Selection in Inidirect Regression by Residual Based Bootstrap”, Statist. Sinica (2018) (in press).

  5. N. Bissantz, T. Hohage, A. Munk, and F. Ruymgaart, “Convergence Rates of General Regularization Methods for Statistical Inverse Problems”, SIAM J. Numerical Anal. 45, 2610–2636 (2007).

    MathSciNet  Article  MATH  Google Scholar 

  6. N. Bissantz and H. Holzmann, “Asymptotics for Spectral Regularization Estimators in Statistical Inverse Problems”, Comput. Statist. 28, 435–453 (2013).

    MathSciNet  Article  MATH  Google Scholar 

  7. G. Blanchard and N. Mücke, “Optimal Rates for Regularization of Statistical Inverse Learning Problems”, Foundations Comput. Math. 18(4), 971–1013 (2018).

    MathSciNet  Article  MATH  Google Scholar 

  8. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

    Google Scholar 

  9. T. Buzug, Computed Tomography. From Photon Statistics to Modern Cone-Beam CT (Springer-Verlag, Berlin-Heidelberg, 2008).

    Google Scholar 

  10. R. J. Carroll, A. Delaigle, and P. Hall, “Non-parametric Regression Estimation from Data Contaminated by a Mixture of Berkson and Classical Errors”, J. Roy. Statist. Soc.: Ser. B, Statist. Methodol. 69, 859–878 (2007).

    MathSciNet  Article  Google Scholar 

  11. L. Cavalier, “Asymptotically Efficient Estimation in a Problem Related to Tomography”, Math. Methods Statist. 7, 445–456 (1999).

    MathSciNet  MATH  Google Scholar 

  12. L. Cavalier, “Efficient Estimation of a Density in a Problem of Tomography”, Ann. Statist. 28, 630–647 (2000).

    MathSciNet  Article  MATH  Google Scholar 

  13. L. Cavalier, “Nonparametric Statistical Inverse Problems”, Inverse Problems 24, 034004, 19 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  14. L. Cavalier and Yu. Golubev, “Risk Hull Method and Regularization by Projections of Ill-posed Inverse Problems”, Ann. Statist. 34, 1653–1677 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  15. L. Cavalier and A. Tsybakov, “Sharp Adaptation for Inverse Problems with Random Noise”, Probab. Theory and Related Fields 123(3), 323–354 (2002).

    MathSciNet  Article  MATH  Google Scholar 

  16. B. Colling and I. van Keilegom, “Goodness-of-Fit Tests in Semiparametric Transformation Models”, TEST 25(2), 291–308 (2016).

    MathSciNet  Article  MATH  Google Scholar 

  17. S. R. Deans, The Radon Transform and some of its Applications (Wiley, New York, 1983).

    MATH  Google Scholar 

  18. A. Delaigle and P. Hall, “Estimation of Observation-Error Variance in Errors-in-Variables Regression”, Statist. Sinica 21, 103–1063 (2011).

    MathSciNet  MATH  Google Scholar 

  19. A. Delaigle, P. Hall, and F. Jamshidi, “Confidence Bands in Non-parametric Errors-in-Variables Regression”, J. Roy. Statist. Soc.: Ser. B, Statist. Methodology 77 (2014).

  20. H. Dette, N. Neumeyer, and I. van Keilegom, “A New Test for the Parametric Form of the Variance Function in Non-parametric Regression”, J. Roy. Statist. Soc.: Ser. B, Statist. Methodology 69(5), 903–917 (2007).

    MathSciNet  Article  Google Scholar 

  21. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, in Mathematics and its Applications, (Kluwer Academic Publishers Group, Dordrecht, 1996), Vol. 375.

    Google Scholar 

  22. S. Helgason, Integral Geometry and Radon Transforms (Springer, New York, 2011).

    Book  MATH  Google Scholar 

  23. T. Hotz, P. Marnitz, R. Stichtenoth, L. Davies, Z. Kabluchko, and A. Munk, “Locally Adaptive Image Denoising by a Statistical Multiresolution Criterion”, Comput. Statist. Data Anal. 56, 543–558 (2012).

    MathSciNet  MATH  Google Scholar 

  24. A. J. E. M. Janssen, “Zernike Expansion of Derivatives and Laplacians of the Zernike Circle Polynomials”, J. Opt. Soc. Am. A 31(7), 1604–1613 (2014).

    Article  Google Scholar 

  25. I. M. Johnstone and B. W. Silverman, “Speed of Estimation in Positron Emission Tomography and Related Inverse Problems”, Ann. Statist. 18(1), 251–280 (1990).

    MathSciNet  Article  MATH  Google Scholar 

  26. J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems (Springer, Berlin, 2010).

    MATH  Google Scholar 

  27. K. Kato and Y. Sasaki, Uniform Confidence Bands for Nonparametric Errors-in-Variables Regression, arXiv:1702.03377v3 (2017).

  28. G. Kerkyacharian, E. Le Pennec, and D. Picard, “Radon Needlet Thresholding”, Bernoulli 18(2), 391–433 (2012).

    MathSciNet  Article  MATH  Google Scholar 

  29. G. Kerkyacharian, G. Kyriazis, E. Le Pennec, P. Petrushev, and D. Picard, “Inversion of Noisy Radon Transform by SVD Based Needlets”, Appl. Comput. Harmonic Anal. 28(1), 24–45 (2010).

    MathSciNet  Article  MATH  Google Scholar 

  30. E. V. Khmaladze and H. L. Koul, “Goodness-of-Fit Problem for Errors in Nonparametric Regression: Distribution Free Approach”, Ann. Statist. 37(6A), 3165–3185 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  31. A. P. Korostelev and A. B. Tsybakov, Minimax Theory of Image Reconstruction, in Lecture Notes in Statistics (Springer-Verlag, New York, 1993), Vol. 82.

    Google Scholar 

  32. H. L. Koul and W. Song, “Regression Model Checking with Berkson Measurement Errors”, J. Statist. Plann. Inference 138, 1615–1628 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  33. H. L. Koul and W. Song, “Minimum Distance Regression Model Checking with Berkson Measurement Errors”, Ann. Statist. 37, 132–156 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  34. V. Lakshminarayanan and A. Fleck, “Zernike Polynomials: A Guide”, J. Modern Optics 58(7), 545–561 (2011).

    Article  Google Scholar 

  35. B. A. Mair and F. H. Ruymgaart, “Statistical Inverse Estimation in Hilbert Scales”, SIAMJ. Appl. Math. 56(5), 1424–1444 (1996).

    MathSciNet  Article  MATH  Google Scholar 

  36. J. C. Mason and D. Handscomb, Chebyshev Polynomials (A CRC Press Company, London, 2002).

    Book  MATH  Google Scholar 

  37. U. Müller, A. Schick, and W. Wefelmeyer, “Estimating the Error Distribution Function in Semiparametric Additive Regression Models”, J. Statist. Plann. Inference 142, 552–566 (2012).

    MathSciNet  Article  MATH  Google Scholar 

  38. F. Natterer, “Computerized Tomography with Unknown Sources”, SIAM J. Appl. Math. 43(5), 1201–1212 (1983a).

    MathSciNet  Article  MATH  Google Scholar 

  39. F. Natterer, The Mathematics of Computerized Tomography (Teubner, Stuttgart; Wiley, Chichester; 1986).

    MATH  Google Scholar 

  40. F. Natterer and F. Wübbelling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).

    Book  Google Scholar 

  41. S. H. Nawab, A. S. Willsky, and A. V. Oppenheim, Signals and Systems (Prentice Hall, New Jersey, 1996).

    Google Scholar 

  42. N. Neumeyer, Bootstrap Procedures for Empirical Processes of Nonparametric Residuals (Ruhr-Universität, Bochum, 2006), habilitation thesis. https://www.math.uni-hamburg.de/home/neumeyer/habil.ps

    Google Scholar 

  43. N. Neumeyer, “Smooth Residual Bootstrap for Empirical Processes of Non-parametric Regression Residuals”, Scandinavian J. Statist. 36(2), 204–228 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  44. N. Neumeyer and I. van Keilegom, “Estimating the Error Distribution in Nonparametric Multiple Regression with Applications to Model Testing”, J. Multivariate Anal. 101(5), 1067–1078 (2010).

    MathSciNet  Article  MATH  Google Scholar 

  45. J. P. Romano and A. F. Siegel, Counterexamples in Probability and Statistics (Chapman and Hall, New York, 1986).

    MATH  Google Scholar 

  46. S. Saitoh, “Integral Transforms, Reprodücing Kernels and their Applications” (Longman, Harlow, 1997).

    MATH  Google Scholar 

  47. B. W. Silverman, “Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and its Derivatives”, Ann. Statist. 6(1), 177–184 (1978).

    MathSciNet  Article  MATH  Google Scholar 

  48. A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes. With Applications to Statistics, in Springer Series in Statistics (Springer, New York, 1996).

    MATH  Google Scholar 

  49. C. R. Vogel, Computational Methods for Inverse Problems (SIAM, Bozeman, 2002).

    Book  MATH  Google Scholar 

  50. F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrast-methode”, Physica 1(8), 689–704 (1934).

    Article  MATH  Google Scholar 

  51. J. Zhang, Z. Feng, and X. Wang, “A Constructive Hypothesis Test for the Single-Index Models with Two Groups”, Ann. Inst. Statist. Math. 70(5), 1077–1114 (2018).

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Project C1) of the German Research Foundation (DFG) and the Bundesministerium für Bildung und Forschung through the project “MED4D: Dynamic medical imaging: Modeling and analysis of medical data for improved diagnosis, supervision and drug development”.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to T. Kutta, N. Bissantz, J. Chown or H. Dette.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kutta, T., Bissantz, N., Chown, J. et al. The Empirical Process of Residuals from an Inverse Regression. Math. Meth. Stat. 28, 104–126 (2019). https://doi.org/10.3103/S1066530719020029

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530719020029

Keywords

  • indirect regression model
  • inverse problems
  • Radon transform
  • empirical process

AMS 2010 Subject Classification

  • 62G08
  • 62G30
  • 15A29