Density estimation by wavelet thresholding

Annals of Statistics

Volumn 24, Issue 2, 1996, Pages 508-539

  Donoho, David L ^a   Johnstone, Iain M ^a   Kerkyacharian, Gérard ^b   Picard, Dominique ^c  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITÉ DE PICARDIE JULES VERNE   (France)

^c CNRS   (France)

Author keywords

Adaptive estimation; Besov spaces; Density estimation; Minimax estimation; Spatial adaptation; Wavelet orthonormal bases

Indexed keywords

EID: 0030504418     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1032894451     Document Type: Article

References (32)

1. BERGH, J. and LÖFSTRÖM, J. (1976). Interpolation Spaces - An Introduction. Springer, New York.
2. BIRGÉ, L. and MASSART, P. (1996). From model selection to adaptive estimation. In Festschrift for Lucien Le Cam (D. Pollard and G. Yang, eds.). Springer, New York.
3. DAUBECHIES, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
4. DELYON, B. and JUDITSKY, A. (1993). Wavelet estimators, global error measures: revisited. Technical Report 782, Institut de Recherche en Informatique et Systèmes Aléatoires, Campus de Beaulieu.
5. 1 View. Wiley, New York.
6. q-error. Probab. Theory Related Fields 99 277-303.
7. DONOHO, D. L. and JOHNSTONE, I. M. (1996). Minimax estimation via wavelet shrinkage. Unpublished manuscript.
8. DONOHO, D. L., JOHNSTONE, I. M., KERKYACHARIAN, G. and PICARD, D. (1996). Universal near minimaxity of wavelet shrinkage. In Festschrift for Lucien Le Cam (D. Pollard and G. Yang, eds.). Springer, New York.
9. DONOHO, D. L., JOHNSTONE, I. M., KERKYACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: asymptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369.
10. DONOHO, D. L., LIU, R. C. and MACGIBBON, K. B. (1990). Minimax risk over hyperrectangles, and implications. Ann. Statist. 18 1416-1437.
11. DOUKHAN, P. and LEON, J. (1990). Déviation quadratique d'estimateurs d'une densité par projection orthogonale. C. R. Acad. Sci. Paris Sér. I Math. 310 425-430.
12. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
13. FIX, G. and STRANG, G. (1969). A Fourier analysis of the finite element method. Stud. Appl. Math. 48 265-273.
14. FRAZIER, M., JAWERTH, B. and WEISS, G. (1991). Littlewood-Paley Theory and the Study of Function Spaces. Amer. Math. Soc., Providence, RI.
15. HÄRDLE, W., KERKYACHARIAN, G., PICARD, D. and TSYBAKOV, A. (1996). Wavelets and econometric applications. Technical report, Humboldt Univ., Berlin.
16. JOHNSTONE, I., KERKYACHARIAN, G. and PICARD, D. (1992). Estimation d'une densité de probabilité par méthode d'ondelettes. C. R. Acad. Sci. Paris Sér. I Math. 315 211-216.
17. KERKYACHARIAN, G. and PICARD, D. (1992). Density estimation in Besov spaces. Statist. Probab. Lett. 13 15-24.
18. KERKYACHARIAN, G. and PICARD, D. (1993). Density estimation by kernel and wavelet methods: optimality of Besov spaces. Statist. Probab. Lett. 18 327-336.
19. MEYER, Y. (1990). Ondelettes et Opérateurs, I: Ondelettes, II: Opérateurs de Calderón-Zygmund, III: (with R. Coifman), Opérateurs Multilinéaires. Hermann, Paris. (English translation of first volume published by Cambridge Univ. Press.)
20. NEMIROVSKII, A. (1985). Nonparametric estimation of smooth regression function. Izv. Akad. Nauk. SSSR Tekhn. Kibernet. 3 50-60 (in Russian); Soviet J. Comput. Systems Sci. 23 1-11 (1986) (in English).
21. NEMIROVSKII, A. (1985). Nonparametric estimation of smooth regression function. Izv. Akad. Nauk. SSSR Tekhn. Kibernet. 3 50-60 (in Russian); Soviet J. Comput. Systems Sci. 23 1-11 (1986) (in English).
22. NEMIROVSKII, A., POLYAK, B. and TSYBAKOV, A. (1985). Rate of convergence of nonparametric estimates of maximum-likelihood type. Problems Inform. Transmission 21 258-272.
23. NUSSBAUM, M. (1995). Personal communication.
24. PEETRE, J. (1975). New Thoughts on Besov Spaces. Dept. Mathematics, Duke Univ.
25. PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
26. POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York.
27. p of sequences of independent random variables. Israel J. Math. 8 273-303.
28. SAKHANENKO, A. I. (1991). Berry-Esseen type estimates for large deviation probabilities. Siberian Math. J. 32 647-656.
29. SCOTT, D. (1992). Multivariate Density Estimation. Wiley, New York.
30. SILVERMAN, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
31. TRIEBEL, H. (1992). Theory of Function Spaces 2. Birkhäuser, Basel.
32. WALTER, G. (1992). Approximation of the delta function by wavelets. J. Approx. Theory 71 329-343.