Deconvolution with unknown error distribution

Annals of Statistics

Volumn 37, Issue 5 A, 2009, Pages 2301-2323

  Johannes, Jan ^a  

^a UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

Deconvolution; Fourier transform; Kernel estimation; Optimal rate of convergence; Sobolev space; Source condition; Spectral cut off

Indexed keywords

EID: 69049107546     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/08-AOS652     Document Type: Article

References (41)

1. Bigot, J. and VAN BELLEGEM, S. (2006). Log-density deconvolution by wavelet thresholding. Technical report, Univ. catholique de Louvain.
2. BUTUCEA, C. and MATIAS, C. (2005). Minimax estimation of the noise level and of the deconvolution density in a semiparametric deconvolution model. Bernoulli 11 309-340. MR2132729
3. CARRASCO, M. and FLORENS, J.-P. (2002). Spectral method for deconvolving a density. Working Paper No. 138, Univ. Toulouse I, IDEI.
4. CARROLL, R. J. and HALL, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184-1186. MR0997599
5. CAVALIER, L. and HENGARTNER, N. (2005). Adaptive estimation for inverse problems with noisy operators. Inverse Problems 21 1345-1361. MR2158113
6. COMTE, F., RoZENHOLC, Y. and TAUPIN, M.-L. (2006). Penalized contrast estimator for density deconvolution. Canad. J. Statist. 34 431-452. MR2328553 (Pubitemid 44784242)
7. DEVROYE, L. (1989). Consistent deconvolution in density estimation. Canad. J. Statist. 17 235-239. MR1033106
8. DIGGLE, P.J. and HALL, P. (1993). A Fourier approach to nonparametric deconvolution of a density estimate. J. Roy. Statist. Soc. Ser. B 55 523-531. MR1224414
9. EFROMOVICH, S. (1997). Density estimation for the case of supersmooth measurement error. J. Amer. Statist. Assoc. 92 526-535. MR1467846
10. ENGL, H.W., HANKE, M. and NEUBAUER, A. (2000). Regularization of Inverse Problems. Kluwer Academic, Dordrecht. MR1408680
11. FAN, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1 257-1272. MR1126324
12. FAN, J. (1992). Deconvolution with supersmooth distributions. Canad. J. Statist. 20 734-747. MR1183078
13. FAN, J. and KOO, J. Y. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734-747. MR1889978
14. GOLDENSHLUGER, A. (1999). On pointwise adaptive nonparametric deconvolution. Bernoulli 5 907-925. MR1715444
15. GOLDENSHLUGER, A. (2000). Density deconvolution in the circular structural model. J. Multivariate Anal. 81 360-375. MR1906385
16. HALL, P. and QIU, P. (2005). Discrete-transform approach to deconvolution problems. Bio- metrika 92 135-148. MR2158615
17. HALL, P.andYAO, Q. (2003). Inference in components of variance models with low replication. Ann. Statist. 31 414-441. MR1983536
18. HAVILIO, M. (2006). Signal deconvolution based expression-detection and background adjustment for microarray data. J. Comput. Biol. 13 63-80. MR2253541
19. HOROWITZ, J. L. and MARKATOU, M. (1996). Semiparametric estimation of regression models for panel data. Rev. Econom. Stud. 63 145-168. MR1372250
20. KAWATA, T. (1972). Fourier Analysis in Probability Theory. Academic Press, New York. MR0464353
21. KIM, P.T. and KOO, J. Y. (2002). Optimal spherical deconvolution. J. Multivariate Anal. 80 21-42. MR1889831
22. KOO, J.Y. and PARK, B. U. (1996). B-spline deconvolution based on the EM algorithm. J. Statist. Comput. Simulation 54 275-288. MR1701219
23. LEVITT, D. G. (2003). The use of a physiologically based pharmacokinetic model to evaluate deconvolution measurements of systemic absorption. In BMC Clinical Pharmacology 3. BioMed Central. Available at http://www.biomedcentral. com/1472-6904/371.
24. LIU, M.C. and TAYLOR, R. L. (1989). A consistent nonparametric density estimator for the deconvolution problem. Canad. J. Statist. 17 427-438. MR1047309
25. MAIR, B. A. (1994). Tikhonov regularization for finitely and infinitely smoothing operators. SIAMJ. Math. Anal. 25 135-147. MR1257145
26. MAIR, B.A. and RUYMGAART, F. H. (1996). Statistical inverse estimation in Hilbert scales. SIAMJ. Math. Anal. 56 1424-1444. MR1409127
27. MEISTER, A. (2006). Density estimation with normal measurement error with unknown variance. Statist. Sinica 16 195-211. MR2256087
28. MENDELSOHN, J. and RICE, J. (1982). Deconvolution of micro-fluorometric histograms with B -splines. J. Amer. Statist. Assoc. 77 748-753.
29. NAIR, M., PEREVERZEV, S. V. and TAUTENHAHN, U. (2005). Regularization in Hilbert scales under general smoothing conditions. Inverse Problems 21 1851-1869. MR2183654
30. NATTERER, F. (1984). Error bounds for Tikhonov regularization in Hilbert scales. Applicable Anal. 18 29-37. MR0762862
31. NEUMANN, M. H. (1997). On the effect of estimating the error density in nonparametric de-convolution. J. Nonparametr. Statist. 7 307-330. MR1460203
32. NEUMANN, M. H. (2006). Deconvolution from panel data with unknown error distribution. J. Multivariate Anal. 98 1955-1968. MR2396948
33. PARZEN, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065-1076. MR0143282
34. PENSKY, M. and VIDAKOVIC, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033-2053. MR1765627
35. PETROV, V. V. (1995). Limit Theorems ofProbability Theory. Sequences ofIndependent Random Variables, 4th ed. Clarendon Press, Oxford. MR1353441
36. STEFANSKI, L. A. (1990). Rate of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229-235. MR1045189
37. TAUTENHAHN, U. (1996). Error estimates for regularization methods in Hilbert scales. SIAM J. Numer. Anal. 33 2120-2130. MR1427456
38. TESSIER, E. (1995). Analysis and calibration of natural guide star adaptive optics data. In Adaptive Optical Systems and Applications (R. K. Tyson and R. Q. Fugate, eds.) 178-193. Proc. SPIE 2534. Available at arXiv:astro-ph/9601174v1.
39. VAINIKKO, G.M. and VERETENNIKOV, A. Y. (1986). Iteration Procedures in Ill-Posed Problems. Nauka, Moscow (in Russian). MR0859375
40. VAN DER VAART, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press. MR1652247
41. ZHANG, C.-H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806-831. MR1056338