Confidence bands for multivariate and time dependent inverse regression models. (English) Zbl 1388.62113

Summary: 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.


62G08 Nonparametric regression and quantile regression
62G07 Density estimation
62G15 Nonparametric tolerance and confidence regions


Full Text: DOI arXiv Euclid


[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. · Zbl 1186.85001
[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. · Zbl 0297.60020
[4] Bickel, P.J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071-1095. · Zbl 0275.62033
[5] Birke, M., Bissantz, N. and Holzmann, H. (2010). Confidence bands for inverse regression models. Inverse Problems 26 115020, 18. · Zbl 1203.62060
[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. · Zbl 1175.62035
[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. · Zbl 1234.62062
[9] Cavalier, L. (2000). Efficient estimation of a density in a problem of tomography. Ann. Statist. 28 630-647. · Zbl 1105.62331
[10] Cavalier, L. (2008). Nonparametric statistical inverse problems. Inverse Problems 24 034004, 19. · Zbl 1137.62323
[11] Cavalier, L. and Tsybakov, A. (2002). Sharp adaptation for inverse problems with random noise. Probab. Theory Related Fields 123 323-354. · Zbl 1039.62031
[12] Claeskens, G. and Van Keilegom, I. (2003). Bootstrap confidence bands for regression curves and their derivatives. Ann. Statist. 31 1852-1884. · Zbl 1042.62044
[13] Engl, H.W., Hanke, M. and Neubauer, A. (1996). Regularization of Inverse Problems. Mathematics and Its Applications 375 . Dordrecht: Kluwer Academic. · Zbl 0859.65054
[14] Eubank, R.L. and Speckman, P.L. (1993). Confidence bands in nonparametric regression. J. Amer. Statist. Assoc. 88 1287-1301. · Zbl 0792.62030
[15] Folland, G.B. (1984). Real Analysis : Modern Techniques and Their Applications. Pure and Applied Mathematics ( New York ). New York: Wiley. · Zbl 0549.28001
[16] Giné, E. and Nickl, R. (2010). Confidence bands in density estimation. Ann. Statist. 38 1122-1170. · Zbl 1183.62062
[17] Hall, P. (1992). On bootstrap confidence intervals in nonparametric regression. Ann. Statist. 20 695-711. · Zbl 0765.62049
[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. · Zbl 0780.62040
[19] Kaipio, J. and Somersalo, E. (2010). Statistical and Computational Inverse Problems . Berlin: Springer. · Zbl 1068.65022
[20] Khoshnevisan, D. (2002). Multiparameter Processes. Springer Monographs in Mathematics . New York: Springer. An introduction to random fields. · Zbl 1005.60005
[21] Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 201-231. · Zbl 1209.62060
[22] Mair, B.A. and Ruymgaart, F.H. (1996). Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 1424-1444. · Zbl 0864.62020
[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. · Zbl 0930.62038
[24] Neumann, M.H. and Polzehl, J. (1998). Simultaneous bootstrap confidence bands in nonparametric regression. J. Nonparametr. Statist. 9 307-333. · Zbl 0913.62041
[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. · Zbl 1266.26024
[26] Paranjape, S.R. and Park, C. (1973). Laws of iterated logarithm of multiparameter Wiener processes. J. Multivariate Anal. 3 132-136. · Zbl 0278.60053
[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. · Zbl 0776.60045
[29] Saitoh, S. (1997). Integral Transforms , Reproducing Kernels and Their Applications. Pitman Research Notes in Mathematics Series 369 . Harlow: Longman. · Zbl 0891.44001
[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. · Zbl 0909.62043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.