×

Multiscale methods for shape constraints in deconvolution: confidence statements for qualitative features. (English) Zbl 1293.62104

Summary: We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task, although the minimax rates for pointwise estimation are very slow.

MSC:

62G10 Nonparametric hypothesis testing
62G15 Nonparametric tolerance and confidence regions
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Abramowitz, M. and Stegun, I. A., eds. (1992). Handbook of Mathematical Functions with Formulas , Graphs , and Mathematical Tables . Dover Publications Inc., New York. · Zbl 0171.38503
[2] Balabdaoui, F., Bissantz, K., Bissantz, N. and Holzmann, H. (2010). Demonstrating single and multiple currents through the E. coli -SecYEG-pore: Testing for the number of modes of noisy observations. J. Amer. Statist. Assoc. 105 136-146. · Zbl 1397.62446
[3] Bissantz, N., Claeskens, G., Holzmann, H. and Munk, A. (2009). Testing for lack of fit in inverse regression-with applications to biophotonic imaging. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 25-48. · Zbl 1231.62060
[4] 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.
[5] Bissantz, N. and Holzmann, H. (2008). Statistical inference for inverse problems. Inverse Problems 24 034009, 17. · Zbl 1137.62325
[6] Butucea, C. and Tsybakov, A. B. (2007). Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl 52 111-128. · Zbl 1141.62021
[7] Chaudhuri, P. and Marron, J. S. (2000). Scale space view of curve estimation. Ann. Statist. 28 408-428. · Zbl 1106.62318
[8] Davies, P. L., Kovac, A. and Meise, M. (2009). Nonparametric regression, confidence regions and regularization. Ann. Statist. 37 2597-2625. · Zbl 1173.62023
[9] De Angelis, D., Gilks, W. R. and Day, N. E. (1998). Bayesian projection of the acquired immune deficiency syndrome epidemic. J. R. Stat. Soc. Ser. C. Appl. Stat. 47 449-498. · Zbl 0935.62121
[10] Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249-267. · Zbl 1429.62125
[11] Diggle, P. J. and Hall, P. (1993). A Fourier approach to nonparametric deconvolution of a density estimate. J. R. Stat. Soc. Ser. B Stat. Methodol. 55 523-531. · Zbl 0783.62030
[12] Dümbgen, L. and Spokoiny, V. G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124-152. · Zbl 1029.62070
[13] Dümbgen, L. and Walther, G. (2008). Multiscale inference about a density. Ann. Statist. 36 1758-1785. · Zbl 1142.62336
[14] Fan, J. (1991). Asymptotic normality for deconvolution kernel density estimators. Sankhyā Ser. A 53 97-110. · Zbl 0729.62034
[15] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257-1272. · Zbl 0729.62033
[16] Floyd, C. E., Jaszczak, R. J., Greer, K. L. and Coleman, R. E. (1985). Deconvolution of compton scatter in SPECT. J. Nucl. Med. 26 403-408.
[17] Giné, E., Koltchinskii, V. and Sakhanenko, L. (2004). Kernel density estimators: Convergence in distribution for weighted sup-norms. Probab. Theory Related Fields 130 167-198. · Zbl 1048.62041
[18] Giné, E., Koltchinskii, V. and Zinn, J. (2004). Weighted uniform consistency of kernel density estimators. Ann. Probab. 32 2570-2605. · Zbl 1052.62034
[19] Giné, E. and Nickl, R. (2010). Confidence bands in density estimation. Ann. Statist. 38 1122-1170. · Zbl 1183.62062
[20] Golubev, G. K. and Levit, B. Y. (1998). Asymptotically efficient estimation in the Wicksell problem. Ann. Statist. 26 2407-2419. · Zbl 0933.62025
[21] Groeneboom, P. and Jongbloed, G. (1995). Isotonic estimation and rates of convergence in Wicksell’s problem. Ann. Statist. 23 1518-1542. · Zbl 0843.62034
[22] Hasminskii, R. Z. (2004). Lower bounds for the risk of nonparametric estimates of the mode. In Contributions to Statistics , Jaroslav Hajek Memorial Volume (J. Jurechkova, ed.) 91-97. Academia, Prague.
[23] Holzmann, H., Bissantz, N. and Munk, A. (2007). Density testing in a contaminated sample. J. Multivariate Anal. 98 57-75. · Zbl 1102.62045
[24] Hörmander, L. (2007). The Analysis of Linear Partial Differential Operators. III : Pseudo-Differential Operators . Springer, Berlin. · Zbl 1115.35005
[25] Ingster, Y. I., Sapatinas, T. and Suslina, I. A. (2011). Minimax nonparametric testing in a problem related to the Radon transform. Math. Methods Statist. 20 347-364. · Zbl 1308.62092
[26] Johnstone, I. M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 547-573. · Zbl 1046.62039
[27] Kacperski, K., Erlandsson, K., Ben-Haim, S. and Hutton, B. (2011). Iterative deconvolution of simultaneous \(^{99m}\) Tc and \(^{201}\) Tl projection data measured on a CdZnTe-based cardiac SPECT scanner. Phys. Med. Biol. 56 1397-1414.
[28] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204 . Elsevier Science B.V., Amsterdam. · Zbl 1092.45003
[29] Laurent, B., Loubes, J. M. and Marteau, C. (2011). Testing inverse problems: A direct or an indirect problem? J. Statist. Plann. Inference 141 1849-1861. · Zbl 1394.62052
[30] Laurent, B., Loubes, J.-M. and Marteau, C. (2012). Non asymptotic minimax rates of testing in signal detection with heterogeneous variances. Electron. J. Stat. 6 91-122. · Zbl 1334.62085
[31] Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 201-231. · Zbl 1209.62060
[32] Meister, A. (2009). Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics 193 . Springer, Berlin. · Zbl 1178.62028
[33] Meister, A. (2009). On testing for local monotonicity in deconvolution problems. Statist. Probab. Lett. 79 312-319. · Zbl 1155.62369
[34] Nickl, R. and Reiß, M. (2012). A Donsker theorem for Lévy measures. J. Funct. Anal. 263 3306-3332. · Zbl 1310.60056
[35] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033-2053. · Zbl 0962.62030
[36] Rachdi, M. and Sabre, R. (2000). Consistent estimates of the mode of the probability density function in nonparametric deconvolution problems. Statist. Probab. Lett. 47 105-114. · Zbl 0977.62042
[37] Schmidt-Hieber, J., Munk, A. and Dümbgen, L. (2013). Supplement to “Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features.” . · Zbl 1293.62104
[38] Söhl, J. and Trabs, M. (2012). A uniform central limit theorem and efficiency for deconvolution estimators. Preprint. Available at . 1208.0687v1 · Zbl 1295.62034
[39] Wieczorek, B. (2010). On optimal estimation of the mode in nonparametric deconvolution problems. J. Nonparametr. Stat. 22 65-80. · Zbl 1182.62078
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.