{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T01:01:44Z","timestamp":1772758904806,"version":"3.50.1"},"reference-count":33,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2002,5,1]],"date-time":"2002-05-01T00:00:00Z","timestamp":1020211200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["GR\/M10229"],"award-info":[{"award-number":["GR\/M10229"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["AF\/001664"],"award-info":[{"award-number":["AF\/001664"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2002,5,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>We use cumulants to derive Bayesian credible intervals for wavelet regression estimates. The first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions. These powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale. Hence, a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set. Johnson transformations then yield the credible intervals themselves. Simulations show that these intervals have good coverage rates, even when the underlying function is inhomogeneous, where standard methods fail. In the case where the curve is smooth, the performance of our intervals remains competitive with established nonparametric regression methods.<\/jats:p>","DOI":"10.1111\/1467-9868.00332","type":"journal-article","created":{"date-parts":[[2003,5,6]],"date-time":"2003-05-06T15:13:15Z","timestamp":1052233995000},"page":"189-205","source":"Crossref","is-referenced-by-count":19,"title":["Posterior Probability Intervals for Wavelet Thresholding"],"prefix":"10.1093","volume":"64","author":[{"given":"Stuart","family":"Barber","sequence":"first","affiliation":[{"name":"University of Bristol , UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guy P.","family":"Nason","sequence":"additional","affiliation":[{"name":"University of Bristol , UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bernard W.","family":"Silverman","sequence":"additional","affiliation":[{"name":"University of Bristol , UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2002,6,20]]},"reference":[{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/1467-9884.00216","article-title":"Wavelet analysis and its statistical applications","volume":"49","author":"Abramovich","year":"2000","journal-title":"Statistician"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/978-1-4612-0567-8_3","article-title":"Bayesian approach to wavelet decomposition and shrinkage","volume":"141","author":"Abramovich","year":"1999","journal-title":"Lect. Notes Statist."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1111\/1467-9868.00151","article-title":"Wavelet thresholding via a Bayesian approach","volume":"60","author":"Abramovich","year":"1998","journal-title":"J. R. Statist. Soc."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-3424-6","volume-title":"Asymptotic Techniques for Use in Statistics","author":"Barndorff-Nielsen","year":"1989"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1002\/env.3170050303","article-title":"Some river wavelets","volume":"5","author":"Brillinger","year":"1994","journal-title":"Environmetrics"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1080\/10485259608832666","article-title":"Uses of cumulants in wavelets analysis","volume":"6","year":"1996","journal-title":"J. Nonparam. Statist."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1093\/biomet\/83.4.727","article-title":"Understanding WaveShrink: variance and bias estimation","volume":"83","author":"Bruce","year":"1996","journal-title":"Biometrika"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1080\/01621459.1997.10473662","article-title":"Adaptive Bayesian wavelet shrinkage","volume":"92","author":"Chipman","year":"1997","journal-title":"J. Am. Statist. Ass."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/978-1-4612-0567-8_6","article-title":"Prior elicitation in the wavelet domain","volume":"141","author":"Chipman","year":"1999","journal-title":"Lect. Notes Statist."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1111\/1467-9868.00257","article-title":"Flexible empirical Bayes estimation for wavelets","volume":"62","author":"Clyde","year":"2000","journal-title":"J. R. Statist. Soc."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1093\/biomet\/85.2.391","article-title":"Multiple shrinkage and subset selection in wavelets","volume":"85","author":"Clyde","year":"1998","journal-title":"Biometrika"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1006\/acha.1993.1005","article-title":"Wavelets on the interval and fast wavelet transforms","volume":"1","author":"Cohen","year":"1993","journal-title":"Appl. Comput. Harm. Anal."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1109\/78.668544","article-title":"Wavelet-based statistical signal processing using hidden Markov models","volume":"46","author":"Crouse","year":"1998","journal-title":"IEEE Trans. Signal Process."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970104","volume-title":"Ten Lectures on Wavelets","author":"Daubechies","year":"1992"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1093\/biomet\/81.3.425","article-title":"Ideal spatial adaptation by wavelet shrinkage","volume":"81","author":"Donoho","year":"1994","journal-title":"Biometrika"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"1200","DOI":"10.1080\/01621459.1995.10476626","article-title":"Adapting to unknown smoothness via wavelet shrinkage","volume":"90","year":"1995","journal-title":"J. Am. Statist. Ass."},{"key":"2023040108504436400_","volume-title":"Local Polynomial Modelling and Its Applications","author":"Fan","year":"1996"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-4473-3","volume-title":"Nonparametric Regression and Generalized Linear Models: a Roughness Penalty Approach","author":"Green","year":"1994"},{"key":"2023040108504436400_","volume-title":"PhD Thesis","author":"Herrick","year":"2000"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"190","DOI":"10.2307\/2346693","article-title":"Algorithm AS 100: Normal\u2013Johnson and Johnson\u2013Normal transformations","volume":"25","author":"Hill","year":"1976","journal-title":"Appl. Statist."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"180","DOI":"10.2307\/2346692","article-title":"Algorithm AS 99: Fitting Johnson curves by moments","volume":"25","author":"Hill","year":"1976","journal-title":"Appl. Statist."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1093\/oso\/9780198504856.003.0037","volume-title":"Bayesian Statistics 6","author":"Holmes","year":"1999"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1093\/biomet\/36.1-2.149","article-title":"Systems of frequency curves generated by methods of translation","volume":"36","author":"Johnson","year":"1949","journal-title":"Biometrika"},{"key":"2023040108504436400_","volume-title":"Continuous Univariate Distributions","author":"Johnson","year":"1994"},{"key":"2023040108504436400_","first-page":"69","article-title":"Multiresolution approximations and wavelet orthonormal bases of L2(R)","volume":"315","author":"Mallat","year":"1989","journal-title":"Trans. Am. Math. Soc."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1109\/18.761332","article-title":"Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors","volume":"45","author":"Moulin","year":"1999","journal-title":"IEEE Trans. Inform. Theory"},{"key":"2023040108504436400_","volume-title":"Wavethresh3 Software","author":"Nason","year":"1998"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1080\/10618600.1994.10474637","article-title":"The discrete wavelet transform in S","volume":"3","author":"Nason","year":"1994","journal-title":"J. Comput. Graph. Statist."},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1214\/aos\/1016120374","article-title":"Adaptive confidence interval for pointwise curve estimation","volume":"28","author":"Picard","year":"2000","journal-title":"Ann. Statist."},{"key":"2023040108504436400_","volume-title":"Kendall's Advanced Theory of Statistics","author":"Stuart","year":"1994"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1080\/01621459.1998.10474099","article-title":"Nonlinear wavelet shrinkage with Bayes rules and Bayes factors","volume":"93","author":"Vidakovic","year":"1998","journal-title":"J. Am. Statist. Ass."},{"key":"2023040108504436400_","volume-title":"Statistical Modeling by Wavelets","year":"1999"},{"key":"2023040108504436400_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-4493-1","volume-title":"Kernel Smoothing","author":"Wand","year":"1995"}],"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%2F1467-9868.00332","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2F1467-9868.00332","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/64\/2\/189\/49721726\/jrsssb_64_2_189.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/64\/2\/189\/49721726\/jrsssb_64_2_189.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T17:47:57Z","timestamp":1733766477000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/64\/2\/189\/7098416"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,5,1]]},"references-count":33,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2002,6,20]]},"published-print":{"date-parts":[[2002,5,1]]}},"URL":"https:\/\/doi.org\/10.1111\/1467-9868.00332","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2002,5]]},"published":{"date-parts":[[2002,5,1]]}}}