{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T04:35:33Z","timestamp":1782275733714,"version":"3.54.5"},"reference-count":37,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2006,7,19]],"date-time":"2006-07-19T00:00:00Z","timestamp":1153267200000},"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","id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2006,9,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>We propose a new \u2018Haar\u2013Fisz\u2019 technique for estimating the time-varying, piecewise constant local variance of a locally stationary Gaussian time series. We apply our technique to the estimation of the spectral structure in the locally stationary wavelet model. Our method combines Haar wavelets and the variance stabilizing Fisz transform. The resulting estimator is mean square consistent, rapidly computable and easy to implement, and performs well in practice. We also introduce the \u2018Haar\u2013Fisz transform\u2019, a device for stabilizing the variance of scaled \u03c72-data and bringing their distribution close to Gaussianity.<\/jats:p>","DOI":"10.1111\/j.1467-9868.2006.00558.x","type":"journal-article","created":{"date-parts":[[2006,7,19]],"date-time":"2006-07-19T16:08:38Z","timestamp":1153325318000},"page":"611-634","source":"Crossref","is-referenced-by-count":44,"title":["Haar\u2013Fisz Estimation of Evolutionary Wavelet Spectra"],"prefix":"10.1093","volume":"68","author":[{"given":"Piotr","family":"Fryzlewicz","sequence":"first","affiliation":[{"name":"University of Bristol , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guy P.","family":"Nason","sequence":"additional","affiliation":[{"name":"University of Bristol , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2006,7,19]]},"reference":[{"key":"2023040708373145400_","first-page":"1154","article-title":"Some extensions in the evolutionary spectral analysis of a stochastic process","volume":"16","author":"Battaglia","year":"1979","journal-title":"Boll. Un. Mat. Ital. B"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1718-3","volume-title":"Nonparametric Statistics for Stochastic Processes","author":"Bosq","year":"1998","edition":"2nd"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0304-4076(01)00069-0","article-title":"Forecasting multifractal volatility","volume":"105","author":"Calvet","year":"2001","journal-title":"J. Econometr."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/10485259808832752","article-title":"A wavelet analysis for time series","volume":"10","author":"Chiann","year":"1999","journal-title":"J. Nonparam. Statist."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/0304-4149(95)00090-9","article-title":"On the Kullback-Leibler information divergence of locally stationary processes","volume":"62","author":"Dahlhaus","year":"1996","journal-title":"Stoch. Processes Appl."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","DOI":"10.1214\/009053606000000867","article-title":"Nonparametric quasi maximum likelihood estimation for Gaussian locally stationary processes","author":"Dahlhaus","year":"2006","journal-title":"Ann. Statist."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970104","volume-title":"Ten Lectures on Wavelets","author":"Daubechies","year":"1992"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1198\/016214505000000745","article-title":"Structural break estimation for nonstationary time series models","volume":"101","author":"Davis","year":"2006","journal-title":"J. Am. Statist. Ass."},{"key":"2023040708373145400_","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":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"138","DOI":"10.4064\/cm-3-2-138-146","article-title":"The limiting distribution of a function of two independent random variables and its statistical application","volume":"3","author":"Fisz","year":"1955","journal-title":"Colloq. Math."},{"key":"2023040708373145400_","first-page":"503","article-title":"Modelling and forecasting financial log-returns as locally stationary wavelet processes","volume":"32","author":"Fryzlewicz","year":"2005","journal-title":"J. Appl. Statist."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1198\/106186004X2697","article-title":"A Haar-Fisz algorithm for Poisson intensity estimation","volume":"13","author":"Fryzlewicz","year":"2004","journal-title":"J. Comput. Graph. Statist."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","DOI":"10.1093\/biomet\/93.3.687","article-title":"A Haar\u2013Fisz technique for locally stationary volatility estimation","volume":"93","author":"Fryzlewicz","year":"2006","journal-title":"Biometrika"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1007\/BF02523391","article-title":"Forecasting non-stationary time series by wavelet process modelling","volume":"55","author":"Fryzlewicz","year":"2003","journal-title":"Ann. Inst. Statist. Math."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1111\/1467-9892.00048","article-title":"Choice of thresholds for wavelet shrinkage estimate of the spectrum","volume":"18","author":"Gao","year":"1997","journal-title":"J. Time Ser. Anal."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0167-7152(94)90019-1","article-title":"Moments of the ratio of two dependent quadratic forms","volume":"20","author":"Ghazal","year":"1994","journal-title":"Statist. Probab. Lett."},{"key":"2023040708373145400_","volume-title":"Distributions in Statistics: Continuous Univariate Distributions","author":"Johnson","year":"1970"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/0167-7152(87)90086-1","article-title":"On moments of ratios of quadratic forms in normal variables","volume":"6","author":"Jones","year":"1987","journal-title":"Statist. Probab. Lett."},{"key":"2023040708373145400_","author":"Kim","year":"1998"},{"key":"2023040708373145400_","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":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aos\/1030563977","article-title":"Adaptive covariance estimation of locally stationary processes","volume":"26","author":"Mallat","year":"1998","journal-title":"Ann. Statist."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1111\/j.1467-9892.1989.tb00014.x","article-title":"Contributions to evolutionary spectral theory","volume":"10","author":"M\u00e9lard","year":"1989","journal-title":"J. Time Ser. Anal."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"3126","DOI":"10.1109\/78.330372","article-title":"Wavelet thresholding techniques for power spectrum estimation","volume":"42","author":"Moulin","year":"1994","journal-title":"IEEE Trans. Signal Process."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1111\/1467-9868.00231","article-title":"Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum","volume":"62","author":"Nason","year":"2000","journal-title":"J. R. Statist. Soc. B"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/978-1-4612-2544-7_17","article-title":"The stationary wavelet transform and some statistical applications","volume":"103","author":"Nason","year":"1995","journal-title":"Lect. Notes Statist."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/978-1-4612-2544-7_18","article-title":"Wavelet thresholding: beyond the Gaussian iid situation","volume":"103","author":"Neumann","year":"1995","journal-title":"Lect. Notes Statist."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1023\/A:1016130108440","article-title":"The SLEX model of a non-stationary random process","volume":"54","author":"Ombao","year":"2002","journal-title":"Ann. Inst. Statist. Math."},{"key":"2023040708373145400_","volume-title":"Bayesian decision theoretic scale adaptive estimation of a log-spectral density","author":"Pensky","year":"2004"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1111\/j.2517-6161.1965.tb01488.x","article-title":"Evolutionary spectra and non-stationary processes (with discussion)","volume":"27","author":"Priestley","year":"1965","journal-title":"J. R. Statist. Soc. B"},{"key":"2023040708373145400_","volume-title":"Spectral Analysis and Time Series","author":"Priestley","year":"1981"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1111\/1467-9469.00202","article-title":"Non-parametric curve estimation by wavelet thresholding with locally stationary errors","volume":"27","author":"Von Sachs","year":"2000","journal-title":"Scand. J. Statist."},{"key":"2023040708373145400_","first-page":"231","article-title":"Some statistical applications for locally stationary processes","volume":"56","author":"Sakiyama","year":"2002","journal-title":"Sci. Math. Jpn"},{"key":"2023040708373145400_","first-page":"265","article-title":"The evolutionary spectra of a harmonizable process","volume":"9","author":"Swift","year":"2000","journal-title":"J. Appl. Statist. Sci."},{"key":"2023040708373145400_","volume-title":"Locally adaptive estimation of evolutionary wavelet spectra","author":"Van Bellegem","year":"2003"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","DOI":"10.1002\/9780470317020","volume-title":"Statistical Modeling by Wavelets","author":"Vidakovic","year":"1999"},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1080\/01621459.1980.10477441","article-title":"Automatic smoothing of the log periodogram","volume":"75","author":"Wahba","year":"1980","journal-title":"J. Am. Statist. Ass."},{"key":"2023040708373145400_","doi-asserted-by":"crossref","first-page":"3153","DOI":"10.1109\/78.735293","article-title":"Spectrum estimation by wavelet thresholding of multitaper estimators","volume":"46","author":"Walden","year":"1998","journal-title":"IEEE Trans. Signal Process."}],"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.1467-9868.2006.00558.x","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/68\/4\/611\/49796596\/jrsssb_68_4_611.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/68\/4\/611\/49796596\/jrsssb_68_4_611.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T00:08:18Z","timestamp":1736467698000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/68\/4\/611\/7110627"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,7,19]]},"references-count":37,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2006,7,19]]},"published-print":{"date-parts":[[2006,9,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.1467-9868.2006.00558.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2006,9]]},"published":{"date-parts":[[2006,7,19]]}}}