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Volumn 26, Issue 3, 1998, Pages 879-921

Minimax estimation via wavelet shrinkage

Author keywords

Besov; H lder; Minimax Bayes estimation; Minimax decision theory; Nonlinear estimation; Nonparametric regression; Orthonormal bases of compactly supported wavelets; Renormalization; Sobolev; Triebel spaces; White noise approximation; White noise model

Indexed keywords


EID: 0032334093     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1024691081     Document Type: Article
Times cited : (734)

References (64)
  • 2
    • 0040975585 scopus 로고
    • Minimax estimation of a normal mean subject to doing well at a point
    • (M. H. Rizvi, J. S. Rustagi and D. Siegmund, eds.) Academic Press, New York
    • BICKEL, P. J. (1983). Minimax estimation of a normal mean subject to doing well at a point. In Recent Advances in Statistics (M. H. Rizvi, J. S. Rustagi and D. Siegmund, eds.) 511-528. Academic Press, New York.
    • (1983) Recent Advances in Statistics , pp. 511-528
    • Bickel, P.J.1
  • 5
    • 0030335402 scopus 로고    scopus 로고
    • Asymptotic equivalence of nonparametric regression and white noise
    • BROWN, L. D. and LOW, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398.
    • (1996) Ann. Statist. , vol.24 , pp. 2384-2398
    • Brown, L.D.1    Low, M.G.2
  • 9
    • 84990575058 scopus 로고
    • Orthonormal bases of compactly supported wavelets
    • DAUBECHIES, I. (1988). Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 909-996.
    • (1988) Comm. Pure Appl. Math. , vol.41 , pp. 909-996
    • Daubechies, I.1
  • 12
    • 34249975165 scopus 로고
    • Symmetric iterative interpolation processes
    • DESLAURIERS, G. and DUBUC, S. (1989). Symmetric iterative interpolation processes. Constr. Approx. 5 49-68.
    • (1989) Constr. Approx. , vol.5 , pp. 49-68
    • Deslauriers, G.1    Dubuc, S.2
  • 13
    • 84966241361 scopus 로고
    • Interpolation of Besov spaces
    • DEVORE, R. and POPOV, V. (1988). Interpolation of Besov spaces. Trans. Amer. Math. Soc. 305 397-414.
    • (1988) Trans. Amer. Math. Soc. , vol.305 , pp. 397-414
    • Devore, R.1    Popov, V.2
  • 14
    • 0029307534 scopus 로고
    • De-noising via soft-thresholding
    • DONOHO, D. (1992). De-noising via soft-thresholding. IEEE Trans. Inform. Theory 41 613-627.
    • (1992) IEEE Trans. Inform. Theory , vol.41 , pp. 613-627
    • Donoho, D.1
  • 15
    • 21344480943 scopus 로고
    • Asymptotic minimax risk for sup-norm loss; solution via optimal recovery
    • DONOHO, D. (1994). Asymptotic minimax risk for sup-norm loss; solution via optimal recovery. Probab. Theory Related Fields 99 145-170.
    • (1994) Probab. Theory Related Fields , vol.99 , pp. 145-170
    • Donoho, D.1
  • 16
    • 0029288068 scopus 로고
    • Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition
    • DONOHO, D. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101-126.
    • (1995) Appl. Comput. Harmon. Anal. , vol.2 , pp. 101-126
    • Donoho, D.1
  • 18
    • 0041958932 scopus 로고
    • Ideal spatial adaptation via wavelet shrinkage
    • DONOHO, D. L. and JOHNSTONE, I. M. (1994a). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 425-455.
    • (1994) Biometrika , vol.81 , pp. 425-455
    • Donoho, D.L.1    Johnstone, I.M.2
  • 20
    • 84950459514 scopus 로고
    • Adapting to unknown smoothness via wavelet shrinkage
    • DONOHO, D. L. and JOHNSTONE, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224.
    • (1995) J. Amer. Statist. Assoc. , vol.90 , pp. 1200-1224
    • Donoho, D.L.1    Johnstone, I.M.2
  • 21
    • 0346681114 scopus 로고    scopus 로고
    • Asymptotic minimaxity of wavelet estimators with sampled data
    • Dept. Statistics, Stanford Univ.
    • DONOHO, D. L. and JOHNSTONE, I. M. (1997). Asymptotic minimaxity of wavelet estimators with sampled data. Technical Report, Dept. Statistics, Stanford Univ.
    • (1997) Technical Report
    • Donoho, D.L.1    Johnstone, I.M.2
  • 24
    • 0001014229 scopus 로고
    • Minimax risk over hyperrectangles, and implications
    • DONOHO, D. L., LIU, R. C. and MACGIBBON, K. B. (1990). Minimax risk over hyperrectangles, and implications. Ann. Statist. 18 1416-1437.
    • (1990) Ann. Statist. , vol.18 , pp. 1416-1437
    • Donoho, D.L.1    Liu, R.C.2    Macgibbon, K.B.3
  • 25
    • 38249016845 scopus 로고
    • Minimax quadratic estimation of a quadratic functional
    • DONOHO, D. L. and NUSSBAUM, M. (1990). Minimax quadratic estimation of a quadratic functional. J. Complexity 6 290-323.
    • (1990) J. Complexity , vol.6 , pp. 290-323
    • Donoho, D.L.1    Nussbaum, M.2
  • 26
    • 0022663198 scopus 로고
    • Interpolation through an iterative scheme
    • DUBUC, S. (1986). Interpolation through an iterative scheme. J. Math. Anal. Appl. 114 185-204.
    • (1986) J. Math. Anal. Appl. , vol.114 , pp. 185-204
    • Dubuc, S.1
  • 27
    • 0010052178 scopus 로고
    • Estimation of square-integrable density on the basis of a sequence of observations
    • EFROIMOVICH, S. and PINSKER, M. (1981). Estimation of square-integrable density on the basis of a sequence of observations. Problems Inform. Transmission 17 182-195. (In Russian in Problemy Peredatsii Informatsii 17 50-68.)
    • (1981) Problems Inform. Transmission , vol.17 , pp. 182-195
    • Efroimovich, S.1    Pinsker, M.2
  • 28
    • 0346681094 scopus 로고    scopus 로고
    • In Russian
    • EFROIMOVICH, S. and PINSKER, M. (1981). Estimation of square-integrable density on the basis of a sequence of observations. Problems Inform. Transmission 17 182-195. (In Russian in Problemy Peredatsii Informatsii 17 50-68.)
    • Problemy Peredatsii Informatsii , vol.17 , pp. 50-68
  • 29
    • 0020152606 scopus 로고    scopus 로고
    • Estimation of square-integrable probability density of a random variable
    • EFROIMOVICH, S. and PINSKER, M. (1982). Estimation of square-integrable probability density of a random variable. Problems Inform. Transmission 18 175-189. (In Russian in Problemy Peredatsii Informatsii 18 19-38.)
    • (1982) Problems Inform. Transmission , vol.18 , pp. 175-189
    • Efroimovich, S.1    Pinsker, M.2
  • 30
    • 0020152606 scopus 로고    scopus 로고
    • In Russian
    • EFROIMOVICH, S. and PINSKER, M. (1982). Estimation of square-integrable probability density of a random variable. Problems Inform. Transmission 18 175-189. (In Russian in Problemy Peredatsii Informatsii 18 19-38.)
    • Problemy Peredatsii Informatsii , vol.18 , pp. 19-38
  • 31
    • 0001242228 scopus 로고
    • Banach spaces related to integrable group representations and their atomic decompositions. I
    • FEICHTINGER, H. and GRÖCHENIG, K. (1989). Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86 307-340.
    • (1989) J. Funct. Anal. , vol.86 , pp. 307-340
    • Feichtinger, H.1    Gröchenig, K.2
  • 34
    • 0002319407 scopus 로고
    • A discrete transform and decomposition of distribution spaces
    • FRAZIER, M. and JAWERTH, B. (1990). A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93 34-170.
    • (1990) J. Funct. Anal. , vol.93 , pp. 34-170
    • Frazier, M.1    Jawerth, B.2
  • 38
    • 0009216977 scopus 로고
    • Bounds for the risks of non-parametric regression estimates
    • IBRAGIMOV, I. A. and KHAS'MINSKII, R. Z. (1982). Bounds for the risks of non-parametric regression estimates. Theory Probab. Appl. 27 84-99.
    • (1982) Theory Probab. Appl. , vol.27 , pp. 84-99
    • Ibragimov, I.A.1    Khas'minskii, R.Z.2
  • 39
    • 0001832379 scopus 로고
    • Estimation hölderiennes ponctuelle des fonctions au moyen des coefficients d'ondelettes
    • JAFFARD, S. (1989). Estimation hölderiennes ponctuelle des fonctions au moyen des coefficients d'ondelettes. C. R. Acad. Sci. Paris Sér. I Math. 308 79-81.
    • (1989) C. R. Acad. Sci. Paris Sér. I Math. , vol.308 , pp. 79-81
    • Jaffard, S.1
  • 40
    • 0347311799 scopus 로고
    • Minimax Bayes, asymptotic minimax and sparse wavelet priors
    • (S. Gupta and J. Berger, eds.) Springer, New York
    • JOHNSTONE, I. M. (1994). Minimax Bayes, asymptotic minimax and sparse wavelet priors. In Statistical Decision Theory and Related Topics V (S. Gupta and J. Berger, eds.) 303-326. Springer, New York.
    • (1994) Statistical Decision Theory and Related Topics V , pp. 303-326
    • Johnstone, I.M.1
  • 41
    • 0000263131 scopus 로고
    • Estimation d'une densité de probabilité par méthode d'ondelettes
    • JOHNSTONE, I., KERKYACHARIAN, G. and PICARD, D. (1992). Estimation d'une densité de probabilité par méthode d'ondelettes. C. R. Acad. Sci. Paris Ser. A 315 211-216.
    • (1992) C. R. Acad. Sci. Paris Ser. A , vol.315 , pp. 211-216
    • Johnstone, I.1    Kerkyacharian, G.2    Picard, D.3
  • 46
    • 0031504935 scopus 로고    scopus 로고
    • Optimal spatial adaptation to inhomogeneous smoothness; an approach based on kernel estimates with variable bandwidth selectors
    • LEPSKI, O., MAMMEN, E. and SPOKOINY, V. (1997). Optimal spatial adaptation to inhomogeneous smoothness; an approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 929-947.
    • (1997) Ann. Statist. , vol.25 , pp. 929-947
    • Lepski, O.1    Mammen, E.2    Spokoiny, V.3
  • 48
    • 0024904699 scopus 로고
    • Multifrequency channel decompositions of images and wavelet models
    • MALLAT, S. G. (1989b). Multifrequency channel decompositions of images and wavelet models. IEEE Trans. on Acoust. Signal Speech Process. 37 2091-2110.
    • (1989) IEEE Trans. on Acoust. Signal Speech Process. , vol.37 , pp. 2091-2110
    • Mallat, S.G.1
  • 50
    • 0004316322 scopus 로고
    • Hermann, Paris. (English translation published by Cambridge Univ. Press.)
    • MEYER, Y. (1990a). Ondelettes et Opérateurs I. Hermann, Paris. (English translation published by Cambridge Univ. Press.)
    • (1990) Ondelettes et Opérateurs I
    • Meyer, Y.1
  • 52
    • 0000075686 scopus 로고
    • Ondelettes sur l'intervalle
    • MEYER, Y. (1991). Ondelettes sur l'intervalle. Rev. Mat. Iberoamericana 7 115-133.
    • (1991) Rev. Mat. Iberoamericana , vol.7 , pp. 115-133
    • Meyer, Y.1
  • 53
    • 0000858754 scopus 로고
    • Variable bandwidth kernel estimators of regression curves
    • MÜLLER, H.-G. and STADTMULLER, U. (1987). Variable bandwidth kernel estimators of regression curves. Ann. Statist. 15 182-201.
    • (1987) Ann. Statist. , vol.15 , pp. 182-201
    • Müller, H.-G.1    Stadtmuller, U.2
  • 54
    • 0006305792 scopus 로고
    • Nonparametric estimation of smooth regression functions
    • in Russian
    • NEMIROVSKII, A. (1985). Nonparametric estimation of smooth regression functions. Izv. Akad. Nauk. SSR Teckhn. Kibernet. 3 50-60 (in Russian). J. Comput. System Sci. 23 1-11 (1986) (in English).
    • (1985) Izv. Akad. Nauk. SSR Teckhn. Kibernet. , vol.3 , pp. 50-60
    • Nemirovskii, A.1
  • 55
    • 0347941948 scopus 로고
    • in English
    • NEMIROVSKII, A. (1985). Nonparametric estimation of smooth regression functions. Izv. Akad. Nauk. SSR Teckhn. Kibernet. 3 50-60 (in Russian). J. Comput. System Sci. 23 1-11 (1986) (in English).
    • (1986) J. Comput. System Sci. , vol.23 , pp. 1-11
  • 56
    • 0022140746 scopus 로고
    • Rate of convergence of nonpararnetric estimates of maximum-likelihood type
    • NEMIROVSKII, A., POLYAK, B. and TSYBAKOV, A. (1985). Rate of convergence of nonpararnetric estimates of maximum-likelihood type. Problems Inform. Transmission 21 258-272.
    • (1985) Problems Inform. Transmission , vol.21 , pp. 258-272
    • Nemirovskii, A.1    Polyak, B.2    Tsybakov, A.3
  • 59
    • 0000056062 scopus 로고
    • Optimal filtering of square integrable signals in Gaussian white noise
    • PINSKER, M. (1980). Optimal filtering of square integrable signals in Gaussian white noise. Problems Inform. Transmission 16 120-133. (In Russian in Problemy Peredatsii Informatsii 16 52-68.)
    • (1980) Problems Inform. Transmission , vol.16 , pp. 120-133
    • Pinsker, M.1
  • 60
    • 0347311795 scopus 로고    scopus 로고
    • In Russian
    • PINSKER, M. (1980). Optimal filtering of square integrable signals in Gaussian white noise. Problems Inform. Transmission 16 120-133. (In Russian in Problemy Peredatsii Informatsii 16 52-68.)
    • Problemy Peredatsii Informatsii , vol.16 , pp. 52-68
  • 61
    • 0000935894 scopus 로고
    • Spline smoothing and optimal rates of convergence in nonparametric regression models
    • SPECKMAN, P. (1985). Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Statist. 13 970-983.
    • (1985) Ann. Statist. , vol.13 , pp. 970-983
    • Speckman, P.1
  • 62
    • 0000439527 scopus 로고
    • Optimal global rates of convergence for nonparametric estimators
    • STONE, C. (1982). Optimal global rates of convergence for nonparametric estimators. Ann. Statist. 10 1040-1053.
    • (1982) Ann. Statist. , vol.10 , pp. 1040-1053
    • Stone, C.1


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