A note on the wavelet oracle
Identifieur interne : 002632 ( Istex/Corpus ); précédent : 002631; suivant : 002633A note on the wavelet oracle
Auteurs : Peter Hall ; Gérard Kerkyacharian ; Dominique PicardSource :
- Statistics and Probability Letters [ 0167-7152 ] ; 1999.
English descriptors
- KwdEn :
- Block thresholding, Cient, Convergence rate, Donoho, Elsevier science, Empirical wavelet expansion, Estimator, Individual functions, Logarithmic factor, Nonparametric regression, Optimal rate, Oracle, Previous paragraph, Same performance, Statistics probability letters, Target function, Thresholding, Unknown smoothness, Variance, Wavelet, Wavelet expansion, Wavelet methods, Wavelet shrinkage.
- Teeft :
- Block thresholding, Cient, Convergence rate, Donoho, Elsevier science, Empirical wavelet expansion, Estimator, Individual functions, Logarithmic factor, Nonparametric regression, Optimal rate, Oracle, Previous paragraph, Same performance, Statistics probability letters, Target function, Thresholding, Unknown smoothness, Variance, Wavelet, Wavelet expansion, Wavelet methods, Wavelet shrinkage.
Abstract
Abstract: The extent to which wavelet function estimators achieve benchmark levels of performance is sometimes described in terms of our ability to interpret a mythical oracle, who has access to the “truth” about the target function. Since he is so wise, he is able to threshold in an optimal manner – that is, to include a term in the empirical wavelet expansion if and only if the square of the corresponding true coefficient is larger than the variance of its estimate. In this note we show that if thresholding is performed in blocks then, for piecewise-smooth functions, we can achieve the same first-order mean-square performance as the oracle, right down to the constant factor, for fixed, piecewise-smooth functions.
Url:
DOI: 10.1016/S0167-7152(98)00282-X
Links to Exploration step
ISTEX:CD632646B10CB8BD45F807E7C0B570B4F7BEA4A4Le document en format XML
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Australia</mods:affiliation></affiliation><affiliation><mods:affiliation>Faculté Mathematiques et Informatiques, Université de Picardie, 33 rue Saint-Leu, 80039 Amiens, Cedex 01, France</mods:affiliation></affiliation></author><author><name sortKey="Picard, Dominique" sort="Picard, Dominique" uniqKey="Picard D" first="Dominique" last="Picard">Dominique Picard</name><affiliation><mods:affiliation>Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia</mods:affiliation></affiliation><affiliation><mods:affiliation>Département de Mathematiques, Université de Paris VII, 75251 Paris, Cedex 05, France</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Statistics and Probability Letters</title><title level="j" type="abbrev">STAPRO</title><idno type="ISSN">0167-7152</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1999">1999</date><biblScope 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methods</term><term>Wavelet shrinkage</term></keywords><keywords scheme="Teeft" xml:lang="en"><term>Block thresholding</term><term>Cient</term><term>Convergence rate</term><term>Donoho</term><term>Elsevier science</term><term>Empirical wavelet expansion</term><term>Estimator</term><term>Individual functions</term><term>Logarithmic factor</term><term>Nonparametric regression</term><term>Optimal rate</term><term>Oracle</term><term>Previous paragraph</term><term>Same performance</term><term>Statistics probability letters</term><term>Target function</term><term>Thresholding</term><term>Unknown smoothness</term><term>Variance</term><term>Wavelet</term><term>Wavelet expansion</term><term>Wavelet methods</term><term>Wavelet shrinkage</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: The extent to which wavelet function estimators achieve benchmark levels of performance is 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Since he is so wise, he is able to threshold in an optimal manner – that is, to include a term in the empirical wavelet expansion if and only if the square of the corresponding true coefficient is larger than the variance of its estimate. In this note we show that if thresholding is performed in blocks then, for piecewise-smooth functions, we can achieve the same first-order mean-square performance as the oracle, right down to the constant factor, for fixed, piecewise-smooth functions.</div></front></TEI><istex><corpusName>elsevier</corpusName><keywords><teeft><json:string>wavelet</json:string><json:string>estimator</json:string><json:string>donoho</json:string><json:string>thresholding</json:string><json:string>cient</json:string><json:string>statistics probability letters</json:string><json:string>wavelet shrinkage</json:string><json:string>variance</json:string><json:string>empirical wavelet expansion</json:string><json:string>wavelet expansion</json:string><json:string>optimal rate</json:string><json:string>block thresholding</json:string><json:string>oracle</json:string><json:string>target function</json:string><json:string>same performance</json:string><json:string>elsevier science</json:string><json:string>wavelet methods</json:string><json:string>previous paragraph</json:string><json:string>logarithmic factor</json:string><json:string>individual functions</json:string><json:string>nonparametric regression</json:string><json:string>unknown smoothness</json:string><json:string>convergence rate</json:string></teeft></keywords><author><json:item><name>Peter Hall</name><affiliations><json:string>Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia</json:string><json:string>Corresponding author</json:string><json:string>E-mail: halpstat@fac.anu.edu.au</json:string></affiliations></json:item><json:item><name>Gérard Kerkyacharian</name><affiliations><json:string>Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia</json:string><json:string>Faculté Mathematiques et Informatiques, Université de Picardie, 33 rue Saint-Leu, 80039 Amiens, Cedex 01, France</json:string></affiliations></json:item><json:item><name>Dominique Picard</name><affiliations><json:string>Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia</json:string><json:string>Département de Mathematiques, Université de Paris VII, 75251 Paris, Cedex 05, France</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>primary 62G07</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>secondary 62G20</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Adaptivity</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Bias</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Convergence rate</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Local smoothing</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Nonparametric regression</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Smoothing parameter</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Variance</value></json:item></subject><arkIstex>ark:/67375/6H6-58W739Q7-4</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>Full-length article</json:string></originalGenre><abstract>Abstract: The extent to which wavelet function estimators achieve benchmark levels of performance is sometimes described in terms of our ability to interpret a mythical oracle, who has access to the “truth” about the target function. 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(1993,1995)</json:string><json:string>Donoho and Johnstone, 1994</json:string><json:string>P. Hall et al.</json:string><json:string>Hall and Patil (1995)</json:string><json:string>Juditsky (1997)</json:string><json:string>Hall et al. (1998)</json:string><json:string>Donoho and Johnstone (1995)</json:string><json:string>Donoho et al. 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Since he is so wise, he is able to threshold in an optimal manner – that is, to include a term in the empirical wavelet expansion if and only if the square of the corresponding true coefficient is larger than the variance of its estimate. In this note we show that if thresholding is performed in blocks then, for piecewise-smooth functions, we can achieve the same first-order mean-square performance as the oracle, right down to the constant factor, for fixed, piecewise-smooth functions.</p></abstract><textClass><keywords scheme="keyword"><list><head>MSC</head><item><term>primary 62G07</term></item><item><term>secondary 62G20</term></item></list></keywords></textClass><textClass><keywords scheme="keyword"><list><head>Keywords</head><item><term>Adaptivity</term></item><item><term>Bias</term></item><item><term>Convergence rate</term></item><item><term>Local smoothing</term></item><item><term>Nonparametric regression</term></item><item><term>Smoothing parameter</term></item><item><term>Variance</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1998-09-01">Modified</change><change when="1999">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/CD632646B10CB8BD45F807E7C0B570B4F7BEA4A4/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Elsevier, elements deleted: body; tail"><istex:xmlDeclaration>version="1.0" encoding="utf-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//ES//DTD journal article DTD version 4.5.2//EN//XML" URI="art452.dtd" name="istex:docType"></istex:docType><istex:document><converted-article version="4.5.2" docsubtype="fla"><item-info><jid>STAPRO</jid><aid>2684</aid><ce:pii>S0167-7152(98)00282-X</ce:pii><ce:doi>10.1016/S0167-7152(98)00282-X</ce:doi><ce:copyright type="full-transfer" year="1999">Elsevier Science B.V.</ce:copyright></item-info><head><ce:title>A note on the wavelet oracle</ce:title><ce:author-group><ce:author><ce:given-name>Peter</ce:given-name><ce:surname>Hall</ce:surname><ce:cross-ref refid="AFF1"><ce:sup>a</ce:sup></ce:cross-ref><ce:cross-ref refid="CORR1">*</ce:cross-ref><ce:e-address>halpstat@fac.anu.edu.au</ce:e-address></ce:author><ce:author><ce:given-name>Gérard</ce:given-name><ce:surname>Kerkyacharian</ce:surname><ce:cross-ref refid="AFF1"><ce:sup>a</ce:sup></ce:cross-ref><ce:cross-ref refid="AFF2"><ce:sup>b</ce:sup></ce:cross-ref></ce:author><ce:author><ce:given-name>Dominique</ce:given-name><ce:surname>Picard</ce:surname><ce:cross-ref refid="AFF1"><ce:sup>a</ce:sup></ce:cross-ref><ce:cross-ref refid="AFF3"><ce:sup>c</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="AFF1"><ce:label>a</ce:label><ce:textfn>Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia</ce:textfn></ce:affiliation><ce:affiliation id="AFF2"><ce:label>b</ce:label><ce:textfn>Faculté Mathematiques et Informatiques, Université de Picardie, 33 rue Saint-Leu, 80039 Amiens, Cedex 01, France</ce:textfn></ce:affiliation><ce:affiliation id="AFF3"><ce:label>c</ce:label><ce:textfn>Département de Mathematiques, Université de Paris VII, 75251 Paris, Cedex 05, France</ce:textfn></ce:affiliation><ce:correspondence id="CORR1"><ce:label>*</ce:label><ce:text>Corresponding author</ce:text></ce:correspondence></ce:author-group><ce:date-received day="1" month="7" year="1998"></ce:date-received><ce:date-revised day="1" month="9" year="1998"></ce:date-revised><ce:abstract><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para>The extent to which wavelet function estimators achieve benchmark levels of performance is sometimes described in terms of our ability to interpret a mythical oracle, who has access to the “truth” about the target function. Since he is so wise, he is able to threshold in an optimal manner – that is, to include a term in the empirical wavelet expansion if and only if the square of the corresponding true coefficient is larger than the variance of its estimate. In this note we show that if thresholding is performed in blocks then, for piecewise-smooth functions, we can achieve the same first-order mean-square performance as the oracle, right down to the constant factor, for fixed, piecewise-smooth functions.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:keywords class="msc"><ce:section-title>MSC</ce:section-title><ce:keyword><ce:text>primary 62G07</ce:text></ce:keyword><ce:keyword><ce:text>secondary 62G20</ce:text></ce:keyword></ce:keywords><ce:keywords class="keyword"><ce:section-title>Keywords</ce:section-title><ce:keyword><ce:text>Adaptivity</ce:text></ce:keyword><ce:keyword><ce:text>Bias</ce:text></ce:keyword><ce:keyword><ce:text>Convergence rate</ce:text></ce:keyword><ce:keyword><ce:text>Local smoothing</ce:text></ce:keyword><ce:keyword><ce:text>Nonparametric regression</ce:text></ce:keyword><ce:keyword><ce:text>Smoothing parameter</ce:text></ce:keyword><ce:keyword><ce:text>Variance</ce:text></ce:keyword></ce:keywords></head></converted-article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>A note on the wavelet oracle</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>A note on the wavelet oracle</title></titleInfo><name type="personal"><namePart type="given">Peter</namePart><namePart type="family">Hall</namePart><affiliation>Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia</affiliation><affiliation>Corresponding author</affiliation><affiliation>E-mail: halpstat@fac.anu.edu.au</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Gérard</namePart><namePart type="family">Kerkyacharian</namePart><affiliation>Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia</affiliation><affiliation>Faculté Mathematiques et Informatiques, Université de Picardie, 33 rue Saint-Leu, 80039 Amiens, Cedex 01, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Dominique</namePart><namePart type="family">Picard</namePart><affiliation>Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia</affiliation><affiliation>Département de Mathematiques, Université de Paris VII, 75251 Paris, Cedex 05, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="Full-length article" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">1999</dateIssued><dateModified encoding="w3cdtf">1998-09-01</dateModified><copyrightDate encoding="w3cdtf">1999</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><abstract lang="en">Abstract: The extent to which wavelet function estimators achieve benchmark levels of performance is sometimes described in terms of our ability to interpret a mythical oracle, who has access to the “truth” about the target function. Since he is so wise, he is able to threshold in an optimal manner – that is, to include a term in the empirical wavelet expansion if and only if the square of the corresponding true coefficient is larger than the variance of its estimate. In this note we show that if thresholding is performed in blocks then, for piecewise-smooth functions, we can achieve the same first-order mean-square performance as the oracle, right down to the constant factor, for fixed, piecewise-smooth functions.</abstract><subject><genre>MSC</genre><topic>primary 62G07</topic><topic>secondary 62G20</topic></subject><subject><genre>Keywords</genre><topic>Adaptivity</topic><topic>Bias</topic><topic>Convergence rate</topic><topic>Local smoothing</topic><topic>Nonparametric regression</topic><topic>Smoothing parameter</topic><topic>Variance</topic></subject><relatedItem type="host"><titleInfo><title>Statistics and Probability Letters</title></titleInfo><titleInfo type="abbreviated"><title>STAPRO</title></titleInfo><genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">19990715</dateIssued></originInfo><identifier type="ISSN">0167-7152</identifier><identifier type="PII">S0167-7152(00)X0084-3</identifier><part><date>19990715</date><detail type="volume"><number>43</number><caption>vol.</caption></detail><detail type="issue"><number>4</number><caption>no.</caption></detail><extent unit="issue-pages"><start>325</start><end>434</end></extent><extent unit="pages"><start>415</start><end>420</end></extent></part></relatedItem><identifier type="istex">CD632646B10CB8BD45F807E7C0B570B4F7BEA4A4</identifier><identifier type="ark">ark:/67375/6H6-58W739Q7-4</identifier><identifier type="DOI">10.1016/S0167-7152(98)00282-X</identifier><identifier type="PII">S0167-7152(98)00282-X</identifier><accessCondition type="use and reproduction" contentType="copyright">©1999 Elsevier Science B.V.</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-HKKZVM7B-M">elsevier</recordContentSource><recordOrigin>Elsevier Science B.V., ©1999</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/CD632646B10CB8BD45F807E7C0B570B4F7BEA4A4/metadata/json</uri></json:item></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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