• Bernoulli
  • Volume 21, Number 1 (2015), 144-175.

Confidence bands for multivariate and time dependent inverse regression models

Katharina Proksch, Nicolai Bissantz, and Holger Dette

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Uniform asymptotic confidence bands for a multivariate regression function in an inverse regression model with a convolution-type operator are constructed. The results are derived using strong approximation methods and a limit theorem for the supremum of a stationary Gaussian field over an increasing system of sets. As a particular application, asymptotic confidence bands for a time dependent regression function $f_{t}(x)$ ($x\in\mathbb{R} ^{d}$, $t\in\mathbb{R} $) in a convolution-type inverse regression model are obtained. Finally, we demonstrate the practical feasibility of our proposed methods in a simulation study and an application to the estimation of the luminosity profile of the elliptical galaxy NGC5017. To the best knowledge of the authors, the results presented in this paper are the first which provide uniform confidence bands for multivariate nonparametric function estimation in inverse problems.

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Bernoulli, Volume 21, Number 1 (2015), 144-175.

First available in Project Euclid: 17 March 2015

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confidence bands deconvolution inverse problems multivariate regression nonparametric regression rates of convergence time dependent regression function uniform convergence


Proksch, Katharina; Bissantz, Nicolai; Dette, Holger. Confidence bands for multivariate and time dependent inverse regression models. Bernoulli 21 (2015), no. 1, 144--175. doi:10.3150/13-BEJ563.

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  • [1] Adorf, H. (1995). Hubble space telescope image restauration in its fourth year. Inverse Problems 11 639–653.
  • [2] Bertero, M., Boccacci, P., Desiderà, G. and Vicidomini, G. (2009). Image deblurring with Poisson data: From cells to galaxies. Inverse Problems 25 123006, 26.
  • [3] Bickel, P. and Rosenblatt, M. (1973). Two-dimensional random fields. In Multivariate Analysis, III (Proc. Third Internat. Sympos., Wright State Univ., Dayton, Ohio, 1972) 3–15. New York: Academic Press.
  • [4] Bickel, P.J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
  • [5] Birke, M., Bissantz, N. and Holzmann, H. (2010). Confidence bands for inverse regression models. Inverse Problems 26 115020, 18.
  • [6] Bissantz, N. and Birke, M. (2009). Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators. J. Multivariate Anal. 100 2364–2375.
  • [7] Bissantz, N., Dümbgen, L., Holzmann, H. and Munk, A. (2007). Non-parametric confidence bands in deconvolution density estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 483–506.
  • [8] Bissantz, N., Hohage, T., Munk, A. and Ruymgaart, F. (2007). Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45 2610–2636.
  • [9] Cavalier, L. (2000). Efficient estimation of a density in a problem of tomography. Ann. Statist. 28 630–647.
  • [10] Cavalier, L. (2008). Nonparametric statistical inverse problems. Inverse Problems 24 034004, 19.
  • [11] Cavalier, L. and Tsybakov, A. (2002). Sharp adaptation for inverse problems with random noise. Probab. Theory Related Fields 123 323–354.
  • [12] Claeskens, G. and Van Keilegom, I. (2003). Bootstrap confidence bands for regression curves and their derivatives. Ann. Statist. 31 1852–1884.
  • [13] Engl, H.W., Hanke, M. and Neubauer, A. (1996). Regularization of Inverse Problems. Mathematics and Its Applications 375. Dordrecht: Kluwer Academic.
  • [14] Eubank, R.L. and Speckman, P.L. (1993). Confidence bands in nonparametric regression. J. Amer. Statist. Assoc. 88 1287–1301.
  • [15] Folland, G.B. (1984). Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics (New York). New York: Wiley.
  • [16] Giné, E. and Nickl, R. (2010). Confidence bands in density estimation. Ann. Statist. 38 1122–1170.
  • [17] Hall, P. (1992). On bootstrap confidence intervals in nonparametric regression. Ann. Statist. 20 695–711.
  • [18] Hall, P. (1993). On Edgeworth expansion and bootstrap confidence bands in nonparametric regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 55 291–304.
  • [19] Kaipio, J. and Somersalo, E. (2010). Statistical and Computational Inverse Problems. Berlin: Springer.
  • [20] Khoshnevisan, D. (2002). Multiparameter Processes. Springer Monographs in Mathematics. New York: Springer. An introduction to random fields.
  • [21] Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 201–231.
  • [22] Mair, B.A. and Ruymgaart, F.H. (1996). Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 1424–1444.
  • [23] Neumann, M.H. (1998). Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. Ann. Statist. 26 2014–2048.
  • [24] Neumann, M.H. and Polzehl, J. (1998). Simultaneous bootstrap confidence bands in nonparametric regression. J. Nonparametr. Statist. 9 307–333.
  • [25] Owen, A.B. (2005). Multidimensional variation for quasi-Monte Carlo. In Contemporary Multivariate Analysis and Design of Experiments. Ser. Biostat. 2 49–74. Hackensack, NJ: World Sci. Publ.
  • [26] Paranjape, S.R. and Park, C. (1973). Laws of iterated logarithm of multiparameter Wiener processes. J. Multivariate Anal. 3 132–136.
  • [27] Piterbarg, V.I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs 148. Providence, RI: Amer. Math. Soc. Translated from the Russian by V.V. Piterbarg, Revised by the author.
  • [28] Rio, E. (1993). Strong approximation for set-indexed partial sum processes via KMT constructions. I. Ann. Probab. 21 759–790.
  • [29] Saitoh, S. (1997). Integral Transforms, Reproducing Kernels and Their Applications. Pitman Research Notes in Mathematics Series 369. Harlow: Longman.
  • [30] Smirnov, N.V. (1950). On the construction of confidence regions for the density of distribution of random variables. Doklady Akad. Nauk SSSR (N.S.) 74 189–191.
  • [31] Trujillo, I., Erwin, P., Ramos, A.A. and Graham, A.W. (2004). Evidence for a new elliptical-galaxy paradigm: Sérsic and core galaxies. Astron. J. 127 1917–1942.
  • [32] Xia, Y. (1998). Bias-corrected confidence bands in nonparametric regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 797–811.