A wavelet method for unfolding sphere size distributions

Canadian Journal of Statistics

Volumn 29, Issue 2, 2001, Pages 251-268

  Antoniadis, Anestis ^a   Fan, Jianqing ^b   Gijbels, Irène ^c  

^a UNIV GRENOBLE ALPES   (France)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^c UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Abel integral equation; Adaptive estimation; Besov spaces; Corpuscle problem; Fractional integration; Linear estimators; Wavelet thresholding; Wavelets

Indexed keywords

EID: 0035628673     PISSN: 03195724     EISSN: None     Source Type: Journal    
DOI: 10.2307/3316076     Document Type: Article

References (38)

1. F. Abramovich & B. W. Silverman (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika, 85, 115-129.
2. R. J. Carroll & P. Hall (1988). Optimal rates of convergence for deconvolving a density. Journal of the American Statistical Association, 83, 1184-1186.
3. C. K. Chui (1992). An Introduction to Wavelets. Academic Press, New York.
4. I. Daubechies (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference series in Applied Mathematics, 91, SIAM, Philadelphia, PA.
5. D. L. Donoho (1995). Nonlinear Solution of Linear Inverse Problems by Wavelet-Vaguelette Decomposition. Applied Computational Harmonic Anaysis, 2, 101-126.
6. D. L. Donoho & I. M. Johnstone (1995). Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association, 90, 1200-1224.
7. D. L. Donoho, I. M. Johnstone, G. Kerkyacharian & D. Picard (1996). Density estimation by wavelet thresholding. The Annals of Statistics, 24, 508-539.
8. J. Fan (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. The Annals of Statistics, 21, 600-610.
9. J. Fan & J.-K. Koo (1999). Wavelet deconvolution. Technical report, University of California, Los Angeles, CA.
10. A. Feuerverger & P. Hall (2000). Methods for density estimation in thick-slice versions of Wicksell's problem. Journal of the American Statistical Association, 95, 535-546.
11. C. Gasquet & P. Witomski (1995). Analyse de Fourier et applications: filtrage, calcul numérique, ondelettes. Masson, Paris.
12. G. K. Golubev & B. Y. Levit (1998). Asymptotically efficient estimation in the Wicksell problem. The Annals of Statistics, 26, 2407-2419.
13. R. Gorenflo & S. Vessela (1980). Abel Integral Equations: Analysis and applications. Lecture Notes in Mathematics, 1461, Springer-Verlag, New York.
14. P. Hall & R. L. Smith (1988). The kernel method for unfolding sphere size distributions. Journal of Computational Physics, 74, 409-421.
15. D.-N. Hào, H.-J. Reinhardt & F. Seiffarth (1994). Stable numerical fractional differentiation by mollification. Numerical Functional Analysis and Optimization, 15, 635-659.
16. W. Härdle, G. Kerkyacharian, D. Picard & A. Tsybakov (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics 129, Springer-Verlag, New York.
17. A. W. Hoogendoorn(1992). Estimating the weight undersize distribution for the Wicksell problem. Statistica Neerlandica, 4, 259-282.
18. E. D. Kolaczyk (1996). A wavelet shrinkage approach to tomographic image reconstruction. Journal of the American Statistical Association, 91, 1079-1090.
19. W. C. Krumbein (1935). Thin section mechanical analysis of inducted sediments. Journal of Geology, 43, 482-496.
20. S. Mallat (1989). A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 674-693.
21. S. Mase (1995). Stereological estimation of particle size distributions. Advances in Applied Probability, 27, 350-366.
22. Y. Meyer (1990). Ondelettes et opérateurs, Vol. I and II. Hermann, Paris.
23. D. Nychka, G. Wahba, S. Goldfarb & T. Pugh (1984). Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross sections. Journal of the American Statistical Association, 79, 832-846.
24. F. O'Sullivan (1986). A statistical perspective on ill-posed problems. Statistical Science, 1, 502-518.
25. M. Pensky & B. Vidakovic (1999). Adaptive wavelet estimator for nonparametric density deconvolution. The Annals of Statistics, 27, 2033-2053.
26. D. Pollard (1984). Convergence of Stochastic Processes. Springer-Verlag, New York.
27. B. D. Ripley (1981). Spatial Statistics. Wiley, New York.
28. S. A. Saltykov (1967). The Determination of the Size Distribution of Particles in an Opaque Material from Measurement of the Size Distribution of their Sections. Lecture Notes in Stereology, Springer-Verlag, New York.
29. S. G. Samko, A. A. Kilbas & O. I. Marichev (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York.
30. B. W. Silverman, M. C. Jones, J. D. Wilson & D. Nychka (1990). A smoothed EM approach to indirect estimation problems with particular reference to stereology and emission tomography. Journal of the Royal Statistical Society Series B, 52, 271-324.
31. D. Stoyan, W. S. Kendall & J. Mecke (1995). Stochastic Geometry and its Applications, Second Edition. Wiley, New York.
32. R. Takahashi & M. Sibuya (1996). The maximum size of the planar sections of Random spheres and its application to metallurgy. Annals of the Institute of Statistical Mathematics, 48, 127-144.
33. C. Taylor (1983). A new method for unfolding sphere size distributions. Journal of Microscopy, 132, 57-66.
34. H. Triebel (1983). Theory of Function Spaces II. Monographs in Mathematics, 78. Birkhäuser Verlag, Basel.
35. B. Vidakovic (1999). Statistical Modeling by Wavelets. Wiley, New York.
36. G. G. Walter (1994). Wavelets and Other Orthogonal Systems with Applications. CRC Press, Boca Raton, Florida.
37. J. Weese, E. Korat, D. Maier & J. Honerkamp (1997). Unfolding Sphere Size Distributions with a Density estimator based on Tikhonov regularization. Journal of Computational Physics, 138, 331-353.
38. S. D. Wicksell (1925). The corpuscule problem. A mathematical study of a biometric problem. Biometrika, 17, 84-99.