{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T13:49:45Z","timestamp":1791899385714,"version":"4.3.4"},"reference-count":74,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[1995,7,1]],"date-time":"1995-07-01T00:00:00Z","timestamp":804556800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1995,7,1]]},"abstract":"<jats:title>SUMMARY<\/jats:title>\n                  <jats:p>Much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects\u2014curves, densities, spectral densities, images\u2014from noisy data. A now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems. Unfortunately, the results have rarely moved into practice, for a variety of reasons\u2014among them being similarity to known methods, computational intractability and lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data: translate the empirical wavelet coefficients towards the origin by an amount \u221a(2 log n)\u03c3\/\u221an. The proposal differs from those in current use, is computationally practical and is spatially adaptive; it thus avoids several of the previous objections. Further, the method is nearly minimax both for a wide variety of loss functions\u2014pointwise error, global error measured in Lp-norms, pointwise and global error in estimation of derivatives\u2014and for a wide range of smoothness classes, including standard Holder and Sobolev classes, and bounded variation. This is a much broader near optimality than anything previously proposed: we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity.<\/jats:p>","DOI":"10.1111\/j.2517-6161.1995.tb02032.x","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T01:10:48Z","timestamp":1544145048000},"page":"301-337","source":"Crossref","is-referenced-by-count":1181,"title":["Wavelet Shrinkage: Asymptopia?"],"prefix":"10.1093","volume":"57","author":[{"given":"David L.","family":"Donoho","sequence":"first","affiliation":[{"name":"Stanford University \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Iain M.","family":"Johnstone","sequence":"additional","affiliation":[{"name":"Stanford University \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G\u00e9rard","family":"Kerkyacharian","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Picardie , Amiens , France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dominique","family":"Picard","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Paris VII \u00a0 France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2018,12,5]]},"reference":[{"key":"2024090921150015700_rssb02032-cit-0001","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1093\/biomet\/35.3-4.246","article-title":"The transformation of poisson, binomial and negative-binomial data","volume":"35","author":"Anscombe","year":"1948","journal-title":"Biometrika"},{"key":"2024090921150015700_rssb02032-cit-0002","first-page":"237","article-title":"Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets","volume-title":"Wavelets\u2014a Tutorial in Theory and Applications","author":"Auscher","year":"1992"},{"key":"2024090921150015700_rssb02032-cit-0003","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/BF00532480","article-title":"Approximation dans les \u00e9spaces metriques et th\u00e9orie de l'estimation","volume":"65","author":"Birg\u00e9","year":"1983","journal-title":"Z. Wahrsch. Ver. Geb."},{"key":"2024090921150015700_rssb02032-cit-0004","first-page":"21","article-title":"Nonasymptotic minimax risk for Hellinger balls","volume":"5","author":"Birg\u00e9","year":"1985","journal-title":"Probab. Math. Statist."},{"key":"2024090921150015700_rssb02032-cit-0005","doi-asserted-by":"crossref","first-page":"1532","DOI":"10.1214\/aos\/1176347380","article-title":"The Grenander estimator: A nonasymptotic approach","volume":"17","author":"Birg\u00e9","year":"1989","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0006","volume-title":"CART: Classification and Regression Trees.","author":"Breiman","year":"1983"},{"key":"2024090921150015700_rssb02032-cit-0007","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1080\/00401706.1977.10489521","article-title":"Variable kernel estimates of multivariate densities","volume":"19","author":"Breiman","year":"1977","journal-title":"Technometrics"},{"key":"2024090921150015700_rssb02032-cit-0008","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/BF00535278","article-title":"Estimation des densit\u00e9s: risque minimax","volume":"47","author":"Bretagnolle","year":"1979","journal-title":"Z. Wahrsch. Ver. Geb."},{"key":"2024090921150015700_rssb02032-cit-0009","doi-asserted-by":"crossref","first-page":"1302","DOI":"10.1080\/01621459.1993.10476411","article-title":"Locally adaptive bandwidth choice for kernel regression estimators","volume":"88","author":"Brockmann","year":"1993","journal-title":"J. Am. Statist. Ass."},{"key":"2024090921150015700_rssb02032-cit-0010","volume-title":"A constrained risk inequality with applications to nonparametric functional estimation","author":"Brown","year":"1992"},{"key":"2024090921150015700_rssb02032-cit-0011","first-page":"417","article-title":"Multiresolution analysis, wavelets, and fast algorithms on an interval","volume":"316","author":"Cohen","year":"1992","journal-title":"Compt. Rend. Acad. Sci. Paris A"},{"key":"2024090921150015700_rssb02032-cit-0012","article-title":"Signal processing and compression with wave packets","volume-title":"Proc. Int. Conf. Wavelets, Marseille","author":"Coifman","year":"1989"},{"key":"2024090921150015700_rssb02032-cit-0013","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970104","volume-title":"Ten Lectures on Wavelets.","author":"Daubechies","year":"1992"},{"key":"2024090921150015700_rssb02032-cit-0014","doi-asserted-by":"crossref","DOI":"10.1109\/MILCOM.1992.244110","article-title":"Fast wavelet techniques for near-optimal image processing","volume-title":"Proc. IEEE Military Communications Conf.","author":"DeVore","year":"1992"},{"key":"2024090921150015700_rssb02032-cit-0015","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1090\/S0002-9947-1988-0920166-3","article-title":"Interpolation of Besov spaces","volume":"305","author":"DeVore","year":"1988","journal-title":"Trans. Am. Math. Soc."},{"key":"2024090921150015700_rssb02032-cit-0016","doi-asserted-by":"crossref","first-page":"1390","DOI":"10.1214\/aos\/1176351045","article-title":"One-sided inference about functionals of a density","volume":"16","author":"Donoho","year":"1988","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0017","article-title":"Interpolating wavelet transforms","volume-title":"Technical Report 408.","author":"Donoho","year":"1992"},{"key":"2024090921150015700_rssb02032-cit-0018","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1214\/aos\/1176325367","article-title":"Statistical estimation and optimal recovery","volume":"22","author":"Donoho","year":"1994","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0019","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/BF01199020","article-title":"Asymptotic minimax risk for sup-norm loss; solution via optimal recovery","volume":"99","author":"Donoho","year":"1994","journal-title":"Probab. Theory Reltd Flds"},{"key":"2024090921150015700_rssb02032-cit-0020","article-title":"De-noising via soft-thresholding","volume-title":"IEEE Trans. Inform. Theory","author":"Donoho","year":"1995"},{"key":"2024090921150015700_rssb02032-cit-0021","doi-asserted-by":"crossref","DOI":"10.1006\/acha.1995.1008","article-title":"Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition","volume-title":"Appl. Comput. Harm. Anal.","author":"Donoho","year":"1995"},{"key":"2024090921150015700_rssb02032-cit-0022","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1093\/biomet\/81.3.425","article-title":"Ideal spatial adaptation via wavelet shrinkage","volume":"81","author":"Donoho","year":"1994","journal-title":"Biometrika"},{"key":"2024090921150015700_rssb02032-cit-0023","article-title":"Neo-classical minimax problems, thresholding, and adaptation","volume-title":"Technical Report.","author":"Donoho","year":"1994"},{"key":"2024090921150015700_rssb02032-cit-0024","article-title":"Minimax estimation via wavelet shrinkage","volume-title":"Ann. Statist.","author":"Donoho","year":"1995"},{"key":"2024090921150015700_rssb02032-cit-0025","doi-asserted-by":"crossref","DOI":"10.1080\/01621459.1995.10476626","article-title":"Adapting to unknown smoothness via wavelet shrinkage","volume-title":"J. Am. Statist. Ass.","author":"Donoho","year":"1995"},{"key":"2024090921150015700_rssb02032-cit-0026","article-title":"Wavelet shrinkage: Asymptopia?","volume-title":"Technical Report 419.","author":"Donoho","year":"1993"},{"key":"2024090921150015700_rssb02032-cit-0027","article-title":"Density estimation by wavelet thresholding","volume-title":"Technical Report 426.","author":"Donoho","year":"1993"},{"key":"2024090921150015700_rssb02032-cit-0028","first-page":"668","article-title":"Geometrizing rates of convergence, III","volume":"19","author":"Donoho","year":"1991","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0029","doi-asserted-by":"crossref","first-page":"1416","DOI":"10.1214\/aos\/1176347758","article-title":"Minimax risk over hyperrectangles, and implications","volume":"18","author":"Donoho","year":"1990","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0030","first-page":"182","article-title":"Estimation of square-integrable density on the basis of a sequence of observations","volume":"17","author":"Efromovich","year":"1981","journal-title":"Prob. Inform. Transmssn"},{"key":"2024090921150015700_rssb02032-cit-0031","first-page":"175","article-title":"Estimation of square-integrable probability density of a random variable","volume":"18","author":"Efromovich","year":"1982","journal-title":"Prob. Inform. Transmssn"},{"key":"2024090921150015700_rssb02032-cit-0032","first-page":"58","article-title":"A learning algorithm for nonparametric filtering (in Russian)","volume":"11","author":"Efromovich","year":"1984","journal-title":"Autom. Telem."},{"key":"2024090921150015700_rssb02032-cit-0033","article-title":"Adaption to high spatial inhomogeneity based on wavelets and on local linear smoothing","volume-title":"Technical Report CMA-SR18-93.","author":"Fan","year":"1993"},{"key":"2024090921150015700_rssb02032-cit-0034","article-title":"On local smoothing of nonparametric curve estimators","volume-title":"Technical Report CMA-SR23-93.","author":"Fan","year":"1993"},{"key":"2024090921150015700_rssb02032-cit-0035","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1214\/aoms\/1177698962","article-title":"On the lack of a uniformly consistent sequence of estimates of a density function in certain cases","volume":"38","author":"Farrell","year":"1967","journal-title":"Ann. Math. Statist."},{"key":"2024090921150015700_rssb02032-cit-0036","doi-asserted-by":"crossref","DOI":"10.1090\/cbms\/079","volume-title":"Littlewood-Paley Theory and the Study of Function Spaces.","author":"Frazier","year":"1991"},{"key":"2024090921150015700_rssb02032-cit-0037","first-page":"1","article-title":"Multivariate adaptive regression splines (with discussion)","volume":"19","author":"Friedman","year":"1991","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0038","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1080\/00401706.1989.10488470","article-title":"Flexible parsimonious smoothing and additive modeling","volume":"31","author":"Friedman","year":"1989","journal-title":"Technometrics"},{"key":"2024090921150015700_rssb02032-cit-0039","first-page":"57","article-title":"Adaptive asymptotically minimax estimates of smooth signals","volume":"23","author":"Golubev","year":"1987","journal-title":"Prob. Pered. Inform."},{"key":"2024090921150015700_rssb02032-cit-0040","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1080\/01621459.1980.10477419","article-title":"Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data (with discussion)","volume":"75","author":"Good","year":"1980","journal-title":"J. Am. Statist. Ass."},{"key":"2024090921150015700_rssb02032-cit-0041","article-title":"Formulae for mean integrated squared error of nonlinear wavelet-based density estimators","volume-title":"Technical Report.","author":"Hall","year":"1993"},{"key":"2024090921150015700_rssb02032-cit-0042","article-title":"On wavelet methods for estimating smooth functions","volume-title":"Technical Report.","author":"Hall","year":"1993"},{"key":"2024090921150015700_rssb02032-cit-0043","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1137\/1127008","article-title":"Bounds for the risks of non-parametric regression estimates","volume":"27","author":"Ibragimov","year":"1982","journal-title":"Theory Probab. Applic."},{"key":"2024090921150015700_rssb02032-cit-0044","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/978-1-4612-2618-5_23","article-title":"Minimax bayes, asymptotic minimax and sparse wavelet priors","volume-title":"Statistical Decision Theory and Related Topics, V","author":"Johnstone","year":"1994"},{"key":"2024090921150015700_rssb02032-cit-0045","first-page":"211","article-title":"Estimation d'une densit\u00e9 de probabilit\u00e9 par m\u00e9thode d'ondelettes","volume":"315","author":"Johnstone","year":"1992","journal-title":"Compt. Rend. Acad. Sci. Paris A"},{"key":"2024090921150015700_rssb02032-cit-0046","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1214\/aos\/1176347500","article-title":"Speed of estimation in positron emission tomography and related inverse problems","volume":"18","author":"Johnstone","year":"1990","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0047","article-title":"Asymptotic minimax estimation of regression function in the uniform norm","volume-title":"Theory Probab. Applic.","author":"Korostelev","year":"1993"},{"key":"2024090921150015700_rssb02032-cit-0048","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-5449-2","volume-title":"Extremes and Related Properties of Random Sequences and Processes.","author":"Leadbetter","year":"1983"},{"key":"2024090921150015700_rssb02032-cit-0049","first-page":"1001","article-title":"Sur les coefficients de Fourier des fonctions continues","volume":"285","author":"Leeuw","year":"1977","journal-title":"Compt. Rend. Acad. Sci. Paris A"},{"key":"2024090921150015700_rssb02032-cit-0050","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4171\/rmi\/22","article-title":"Ondelettes et bases hilbertiennes","volume":"2","author":"Lemari\u00e9","year":"1986","journal-title":"Rev. Mat. Iberam."},{"key":"2024090921150015700_rssb02032-cit-0051","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1137\/1135065","article-title":"On a problem of adaptive estimation in gaussian white noise","volume":"35","author":"Lepskii","year":"1991","journal-title":"Theory Probab. Applic."},{"key":"2024090921150015700_rssb02032-cit-0052","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1109\/34.192463","article-title":"A theory for multiresolution signal decomposition: the wavelet representation","volume":"11","author":"Mallat","year":"1989","journal-title":"IEEE Trans. Pattn Anal. Mach. Intell."},{"key":"2024090921150015700_rssb02032-cit-0053","doi-asserted-by":"crossref","first-page":"3397","DOI":"10.1109\/78.258082","article-title":"Matching pursuits with time-frequency dictionaries","volume":"41","author":"Mallat","year":"1993","journal-title":"IEEE Trans. Signal Process."},{"key":"2024090921150015700_rssb02032-cit-0054","volume-title":"Ondelettes et Op\u00e9rateurs: vol. I, ondelettes; vol. II, Op\u00e9rateurs de Calder\u00f3n-Zygmund; vol. III, Op\u00e9rateurs Multilin\u00e9aires.","author":"Meyer","year":"1990"},{"key":"2024090921150015700_rssb02032-cit-0055","doi-asserted-by":"crossref","first-page":"115","DOI":"10.4171\/rmi\/107","article-title":"Ondelettes sur l'intervalle","volume":"7","author":"Meyer","year":"1992","journal-title":"Rev. Mat. Iberam."},{"key":"2024090921150015700_rssb02032-cit-0056","article-title":"Optimal estimation of linear functionals","volume-title":"Technical Report 5729.","author":"Micchelli","year":"1975"},{"key":"2024090921150015700_rssb02032-cit-0057","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/978-1-4684-2388-4_1","article-title":"A survey of optimal recovery","volume-title":"Optimal Estimation in Approximation Theory","author":"Micchelli","year":"1977"},{"key":"2024090921150015700_rssb02032-cit-0058","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1214\/aos\/1176350260","article-title":"Variable bandwidth kernel estimators of regression curves","volume":"15","author":"M\u00fcller","year":"1987","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0059","article-title":"Wavelet regression by cross-validation","volume-title":"Technical Report 447.","author":"Nason","year":"1994"},{"issue":"6","key":"2024090921150015700_rssb02032-cit-0060","first-page":"1","article-title":"Nonparametric estimation of smooth regression function","volume":"23","author":"Nemirovskii","year":"1986","journal-title":"J. Comput. Syst. Sci."},{"key":"2024090921150015700_rssb02032-cit-0061","first-page":"258","article-title":"Rate of convergence of nonparametric estimates of maximum-likelihood type","volume":"21","author":"Nemirovskii","year":"1985","journal-title":"Prob. Inform. Transmssn"},{"key":"2024090921150015700_rssb02032-cit-0062","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1214\/aos\/1176349651","article-title":"Spline smoothing and asymptotic efficiency in l2","volume":"13","author":"Nussbaum","year":"1985","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0063","volume-title":"New Thoughts on Besov Spaces","author":"Peetre","year":"1975"},{"key":"2024090921150015700_rssb02032-cit-0064","first-page":"120","article-title":"Optimal filtering of square integrable signals in gaussian white noise","volume":"16","author":"Pinsker","year":"1980","journal-title":"Prob. Inform. Transmssn"},{"key":"2024090921150015700_rssb02032-cit-0065","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1214\/aos\/1176345399","article-title":"Asymptotically optimum kernels for density estimation at a point","volume":"9","author":"Sacks","year":"1981","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0066","first-page":"1","article-title":"Lower bound for the integral risk of density function estimates","volume-title":"Advances in Soviet Mathematics","author":"Samarov","year":"1992"},{"key":"2024090921150015700_rssb02032-cit-0067","volume-title":"Minimax estimates of linear functionals in a Hilbert space","author":"Speckman","year":"1979"},{"key":"2024090921150015700_rssb02032-cit-0068","doi-asserted-by":"crossref","first-page":"1040","DOI":"10.1214\/aos\/1176345969","article-title":"Optimal global rates of convergence for nonparametric estimators","volume":"10","author":"Stone","year":"1982","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0069","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1214\/aos\/1176348768","article-title":"Variable kernel density estimation","volume":"20","author":"Terrell","year":"1992","journal-title":"Ann. Statist."},{"key":"2024090921150015700_rssb02032-cit-0070","volume-title":"Information-based Complexity.","author":"Traub","year":"1988"},{"key":"2024090921150015700_rssb02032-cit-0071","article-title":"Estimation de densit\u00e9: Analyse multidirnensionelle et m\u00e9thodes d'ondelettes","volume-title":"Th\u00e8se de Doctoral.","author":"Tribouley","year":"1993"},{"key":"2024090921150015700_rssb02032-cit-0072","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-0346-0419-2","volume-title":"Theory of Function Spaces","author":"Triebel","year":"1992"},{"key":"2024090921150015700_rssb02032-cit-0073","first-page":"1","article-title":"A completely automatic French curve","volume":"4","author":"Wahba","year":"1975","journal-title":"Communs Statist."},{"key":"2024090921150015700_rssb02032-cit-0074","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1214\/aoms\/1177729840","article-title":"Minimax estimation of the mean of a normal distribution with known variance","volume":"21","author":"Wolfowitz","year":"1950","journal-title":"Ann. Math. Statist."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fj.2517-6161.1995.tb02032.x","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/57\/2\/301\/49101080\/jrsssb_57_2_301.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/57\/2\/301\/49101080\/jrsssb_57_2_301.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T18:14:28Z","timestamp":1725905668000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/57\/2\/301\/7028285"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,7,1]]},"references-count":74,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,12,5]]},"published-print":{"date-parts":[[1995,7,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.2517-6161.1995.tb02032.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[1995,7]]},"published":{"date-parts":[[1995,7,1]]}}}