Document Zbl 1437.62250

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1437.62250 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">×</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools & Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/1437.62250?"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open — the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span> <a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="an:1437.62250" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span> </span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>  </span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>  </span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>  </span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>  </span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>  </span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>  </span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>  </span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>  </span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>  </span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>  </span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>  </span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>  </span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>  </span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>  </span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>  </span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>  </span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>  </span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a <span class="item">&</span> b </span><span> </span>logical and (default)</a></li> <li><a href="#"><span class="token">a <span class="item">|</span> b </span><span> </span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab   </span><span> </span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>  </span><span> </span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab c<span class="item">"</span></span><span> </span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab c<span class="item">)</span></span><span> </span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki & ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/sun.qiang" title="Author Profile">Sun, Qiang</a>; <a href="/authors/zhou.wen-xin" title="Author Profile">Zhou, Wen-Xin</a>; <a href="/authors/fan.jianqing" title="Author Profile">Fan, Jianqing</a></div> <h2 class="title"> <strong>Adaptive Huber regression.</strong> <i>(English)</i> <a class="label nowrap" href="/1437.62250">Zbl 1437.62250</a> </h2> <div class="source"> <a href="/serials/474" title="Journal Profile">J. Am. Stat. Assoc.</a> <a href="/?q=in%3A451409" title="Articles in this Issue">115, No. 529, 254-265 (2020)</a>. </div> <div class="abstract">Summary: Big data can easily be contaminated by outliers or contain variables with heavy-tailed distributions, which makes many conventional methods inadequate. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. The key observation is that the robustification parameter should adapt to the sample size, dimension and moments for optimal tradeoff between bias and robustness. Our theoretical framework deals with heavy-tailed distributions with bounded $(1+\delta)$th moment for any $\delta>0$. We establish a sharp phase transition for robust estimation of regression parameters in both low and high dimensions: when $\delta\geq1$, the estimator admits a sub-Gaussian-type deviation bound without sub-Gaussian assumptions on the data, while only a slower rate is available in the regime $0<\delta<1$ and the transition is smooth and optimal. In addition, we extend the methodology to allow both heavy-tailed predictors and observation noise. Simulation studies lend further support to the theory. In a genetic study of cancer cell lines that exhibit heavy-tailedness, the proposed methods are shown to be more robust and predictive.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7227129">Cited in <strong>78</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J02" title="MSC2020">62J02</a> </td> <td class="space"> General nonlinear regression </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62F35" title="MSC2020">62F35</a> </td> <td class="space"> Robustness and adaptive procedures (parametric inference) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G32" title="MSC2020">62G32</a> </td> <td class="space"> Statistics of extreme values; tail inference </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aadaptive+Huber+regression">adaptive Huber regression</a>; <a href="/?q=ut%3Abias+and+robustness+tradeoff">bias and robustness tradeoff</a>; <a href="/?q=ut%3Afinite-sample+inference">finite-sample inference</a>; <a href="/?q=ut%3Aheavy-tailed+data">heavy-tailed data</a>; <a href="/?q=ut%3Anonasymptotic+optimality">nonasymptotic optimality</a>; <a href="/?q=ut%3Aphase+transition">phase transition</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1437.62250" data-ciurl="/ci/07227129" data-biburl="/bibtex/07227129.bib" data-amsurl="/amsrefs/07227129.bib" data-xmlurl="/xml/07227129.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07227129.pdf" title="Zbl 1437.62250 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1080/01621459.2018.1543124" aria-label="DOI for “Adaptive Huber regression”" title="10.1080/01621459.2018.1543124">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1706.06991"title="Note: arXiv document may differ from published version">arXiv</a> <a class="btn btn-default btn-xs" type="button" href="https://escholarship.org/uc/item/4sn8g6n9" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Alquier, P.; Cottet, V.; Lecué, G., “Estimation Bounds and Sharp Oracle Inequalities of Regularized Procedures With Lipschitz Loss Functions,”, arXiv no. 1702.01402 (2017)</td> </tr><tr> <td>[2]</td> <td class="space">Belloni, A.; Chernozhukov, V., Penalized Quantile Regression in High-Dimensional Sparse Models,”, The Annals of Statistics, 39, 82-130 (2011) · <a href="/1209.62064" class="nowrap">Zbl 1209.62064</a> · <a href="https://doi.org/10.1214/10-AOS827" class="nowrap">doi:10.1214/10-AOS827</a></td> </tr><tr> <td>[3]</td> <td class="space">Bellec, P. C.; Lecué, G.; Tsybakov, A. B., “Slope Meets Lasso: Improved Oracle Bounds and Optimality,”, The Annals of Statistics, 46, 3603-3642 (2018) · <a href="/1405.62056" class="nowrap">Zbl 1405.62056</a> · <a href="https://doi.org/10.1214/17-AOS1670" class="nowrap">doi:10.1214/17-AOS1670</a></td> </tr><tr> <td>[4]</td> <td class="space">Bickel, P. J.; Ritov, Y.; Tsybakov, A. B., “Simultaneous Analysis of Lasso and Dantzig Selector,”, The Annals of Statistics, 37, 1705-1732 (2009) · <a href="/1173.62022" class="nowrap">Zbl 1173.62022</a> · <a href="https://doi.org/10.1214/08-AOS620" class="nowrap">doi:10.1214/08-AOS620</a></td> </tr><tr> <td>[5]</td> <td class="space">Bogdan, M.; van den Berg, E.; Sabatti, C.; Su, W.; Candès, E. J., “SLOPE—Adaptive Variable Selection via Convex Optimization,”, The Annals of Applied Statistics, 9, 1103-1140 (2015) · <a href="/1454.62212" class="nowrap">Zbl 1454.62212</a> · <a href="https://doi.org/10.1214/15-AOAS842" class="nowrap">doi:10.1214/15-AOAS842</a></td> </tr><tr> <td>[6]</td> <td class="space">Brownlees, C.; Joly, E.; Lugosi, G., “Empirical Risk Minimization for Heavy-Tailed Losses,”, The Annals of Statistics, 43, 2507-2536 (2015) · <a href="/1326.62066" class="nowrap">Zbl 1326.62066</a> · <a href="https://doi.org/10.1214/15-AOS1350" class="nowrap">doi:10.1214/15-AOS1350</a></td> </tr><tr> <td>[7]</td> <td class="space">Bühlmann, P.; van de Geer, S., Statistics for High-Dimensional Data: Methods, Theory and Applications (2011), Heidelberg: Springer, Heidelberg · <a href="/1273.62015" class="nowrap">Zbl 1273.62015</a></td> </tr><tr> <td>[8]</td> <td class="space">Catoni, O., “Challenging the Empirical Mean and Empirical Variance: A Deviation Study,”, Annales de I’Institut Henri Poincaré—Probabilités et Statistiques, 48, 1148-1185 (2012) · <a href="/1282.62070" class="nowrap">Zbl 1282.62070</a> · <a href="https://doi.org/10.1214/11-AIHP454" class="nowrap">doi:10.1214/11-AIHP454</a></td> </tr><tr> <td>[9]</td> <td class="space">Catoni, O., “PAC-Bayesian Bounds for the Gram Matrix and Least Squares Regression With a Random Design,”, arXiv no. 1603.05229 (2016)</td> </tr><tr> <td>[10]</td> <td class="space">Chen, M.; Gao, C.; Ren, Z., “Robust Covariance and Scatter Matrix Estimation Under Huber’s Contamination Model,”, The Annals of Statistics, 46, 1932-1960 (2018) · <a href="/1408.62104" class="nowrap">Zbl 1408.62104</a> · <a href="https://doi.org/10.1214/17-AOS1607" class="nowrap">doi:10.1214/17-AOS1607</a></td> </tr><tr> <td>[11]</td> <td class="space">Cont, R., “Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues,”, Quantitative Finance, 1, 223-236 (2001) · <a href="/1408.62174" class="nowrap">Zbl 1408.62174</a> · <a href="https://doi.org/10.1080/713665670" class="nowrap">doi:10.1080/713665670</a></td> </tr><tr> <td>[12]</td> <td class="space">Delaigle, A.; Hall, P.; Jin, J., “Robustness and Accuracy of Methods for High Dimensional Data Analysis Based on Student’s t-Statistic,”, Journal of the Royal Statistical Society, Series B, 73, 283-301 (2011) · <a href="/1411.62222" class="nowrap">Zbl 1411.62222</a> · <a href="https://doi.org/10.1111/j.1467-9868.2010.00761.x" class="nowrap">doi:10.1111/j.1467-9868.2010.00761.x</a></td> </tr><tr> <td>[13]</td> <td class="space">Devroye, L.; Lerasle, M.; Lugosi, G.; Oliveira, R. I., “Sub-Gaussian Mean Estimators,”, The Annals of Statistics, 44, 2695-2725 (2016) · <a href="/1360.62115" class="nowrap">Zbl 1360.62115</a> · <a href="https://doi.org/10.1214/16-AOS1440" class="nowrap">doi:10.1214/16-AOS1440</a></td> </tr><tr> <td>[14]</td> <td class="space">Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R., “Least Angle Regression,”, The Annals of Statistics, 32, 407-499 (2004) · <a href="/1091.62054" class="nowrap">Zbl 1091.62054</a> · <a href="https://doi.org/10.1214/009053604000000067" class="nowrap">doi:10.1214/009053604000000067</a></td> </tr><tr> <td>[15]</td> <td class="space">Eklund, A.; Nichols, T.; Knutsson, H., “Cluster Failure: Why fMRI Inferences for Spatial Extent Have Inflated False-Positive Rates,”, Proceedings of the National Academy of Sciences of the United States of America, 113, 7900-7905 (2016) · <a href="https://doi.org/10.1073/pnas.1602413113" class="nowrap">doi:10.1073/pnas.1602413113</a></td> </tr><tr> <td>[16]</td> <td class="space">Fan, J.; Fan, Y.; Barut, E., “Adaptive Robust Variable Selection,”, The Annals of Statistics, 42, 324-351 (2014) · <a href="/1296.62144" class="nowrap">Zbl 1296.62144</a> · <a href="https://doi.org/10.1214/13-AOS1191" class="nowrap">doi:10.1214/13-AOS1191</a></td> </tr><tr> <td>[17]</td> <td class="space">Fan, J.; Li, Q.; Wang, Y., “Estimation of High Dimensional Mean Regression in the Absence of Symmetry and Light Tail Assumptions,”, Journal of the Royal Statistical Society, Series B, 79, 247-265 (2017) · <a href="/1414.62178" class="nowrap">Zbl 1414.62178</a> · <a href="https://doi.org/10.1111/rssb.12166" class="nowrap">doi:10.1111/rssb.12166</a></td> </tr><tr> <td>[18]</td> <td class="space">Fan, J.; Li, R., “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties,”, Journal of the American Statistical Association, 96, 1348-1360 (2001) · <a href="/1073.62547" class="nowrap">Zbl 1073.62547</a> · <a href="https://doi.org/10.1198/016214501753382273" class="nowrap">doi:10.1198/016214501753382273</a></td> </tr><tr> <td>[19]</td> <td class="space">Fan, J.; Liu, H.; Sun, Q.; Zhang, T., “I-LAMM for Sparse Learning: Simultaneous Control of Algorithmic Complexity and Statistical Error,”, The Annals of Statistics, 96, 1348-1360 (2018)</td> </tr><tr> <td>[20]</td> <td class="space">Fan, J.; Wang, W.; Zhu, Z., A Shrinkage Principle for Heavy-Tailed Data: High-Dimensional Robust Low-Rank Matrix Recovery, arXiv no. 1603.08315 (2016)</td> </tr><tr> <td>[21]</td> <td class="space">Friedline, J. A.; Garrett, S. H.; Somji, S.; Todd, J. H.; Sens, D. A., “Differential Expression of the MT-1E Gene in Estrogen-Receptor-Positive and -Negative Human Breast Cancer Cell Lines,”, The American Journal of Pathology, 152, 23-27 (1998)</td> </tr><tr> <td>[22]</td> <td class="space">Giulini, I., “Robust PCA and Pairs of Projections in a Hilbert Space,”, Electronic Journal of Statistics, 11, 3903-3926 (2017) · <a href="/1384.62185" class="nowrap">Zbl 1384.62185</a> · <a href="https://doi.org/10.1214/17-EJS1343" class="nowrap">doi:10.1214/17-EJS1343</a></td> </tr><tr> <td>[23]</td> <td class="space">Hastie, T.; Tibshirani, R.; Wainwright, M. J., Statistical Learning With Sparsity: The Lasso and Generalizations (2015), Boca Raton, FL: CRC Press, Boca Raton, FL · <a href="/1319.68003" class="nowrap">Zbl 1319.68003</a></td> </tr><tr> <td>[24]</td> <td class="space">He, X.; Shao, Q.-M, “A General Bahadur Representation of M-Estimators and Its Application to Linear Regression With Nonstochastic Designs,”, The Annals of Statistics, 24, 2608-2630 (1996) · <a href="/0867.62012" class="nowrap">Zbl 0867.62012</a> · <a href="https://doi.org/10.1214/aos/1032181172" class="nowrap">doi:10.1214/aos/1032181172</a></td> </tr><tr> <td>[25]</td> <td class="space">He, X.; Shao, Q.-M, “On Parameters of Increasing Dimensions,”, Journal of Multivariate Analysis, 73, 120-135 (2000) · <a href="/0948.62013" class="nowrap">Zbl 0948.62013</a> · <a href="https://doi.org/10.1006/jmva.1999.1873" class="nowrap">doi:10.1006/jmva.1999.1873</a></td> </tr><tr> <td>[26]</td> <td class="space">Huber, P. J., “Robust Estimation of a Location Parameter,”, The Annals of Mathematical Statistics, 35, 73-101 (1964) · <a href="/0136.39805" class="nowrap">Zbl 0136.39805</a> · <a href="https://doi.org/10.1214/aoms/1177703732" class="nowrap">doi:10.1214/aoms/1177703732</a></td> </tr><tr> <td>[27]</td> <td class="space">Huber, P. J., “Robust Regression: Asymptotics, Conjectures and Monte Carlo,”, The Annals of Statistics, 1, 799-821 (1973) · <a href="/0289.62033" class="nowrap">Zbl 0289.62033</a> · <a href="https://doi.org/10.1214/aos/1176342503" class="nowrap">doi:10.1214/aos/1176342503</a></td> </tr><tr> <td>[28]</td> <td class="space">Koenker, R., Quantile Regression (2005), New York: Cambridge University Press, New York · <a href="/1111.62037" class="nowrap">Zbl 1111.62037</a></td> </tr><tr> <td>[29]</td> <td class="space">Kretzschmar, M., “Transforming Growth Factor-β and Breast Cancer: Transforming Growth Factor-β/Smad Signaling Defects and Cancer,”, Breast Cancer Research, 2, 107-115 (2000)</td> </tr><tr> <td>[30]</td> <td class="space">Landi, A.; Vermeire, J.; Iannucci, V.; Vanderstraeten, H.; Naessens, E.; Bentahir, M.; Verhasselt, B., “Genome-Wide shRNA Screening Identifies Host Factors Involved in Early Endocytic Events for HIV-1-Induced CD4 Down-Regulation,”, Retrovirology, 11, 118-129 (2014) · <a href="https://doi.org/10.1186/PREACCEPT-3552320321395097" class="nowrap">doi:10.1186/PREACCEPT-3552320321395097</a></td> </tr><tr> <td>[31]</td> <td class="space">Lepski, O. V., “Asymptotically Minimax Adaptive Estimation. I. Upper Bounds. Optimally Adaptive Estimates,”, IEEE Transactions on Information Theory, 36, 682-697 (1991) · <a href="/0776.62039" class="nowrap">Zbl 0776.62039</a></td> </tr><tr> <td>[32]</td> <td class="space">Liu, R. Y., “On a Notion of Data Depth Based on Random Simplices,”, The Annals of Statistics, 18, 405-414 (1990) · <a href="/0701.62063" class="nowrap">Zbl 0701.62063</a> · <a href="https://doi.org/10.1214/aos/1176347507" class="nowrap">doi:10.1214/aos/1176347507</a></td> </tr><tr> <td>[33]</td> <td class="space">Liu, R. Y.; Parelius, J. M.; Singh, K., Multivariate Analysis by Data Depth: Descriptive Statistics, Graphics and Inference” (with discussion and a rejoinder by Liu and Singh), The Annals of Statistics, 27, 783-858 (1999) · <a href="/0984.62037" class="nowrap">Zbl 0984.62037</a></td> </tr><tr> <td>[34]</td> <td class="space">Loh, P.; Wainwright, M. J., “Regularized M-Estimators With Nonconvexity: Statistical and Algorithmic Theory for Local Optima,”, Journal of Machine Learning Research, 16, 559-616 (2015) · <a href="/1360.62276" class="nowrap">Zbl 1360.62276</a></td> </tr><tr> <td>[35]</td> <td class="space">Mammen, E., “Asymptotics With Increasing Dimension for Robust Regression With Applications to the Bootstrap,”, The Annals of Statistics, 17, 382-400 (1989) · <a href="/0674.62017" class="nowrap">Zbl 0674.62017</a> · <a href="https://doi.org/10.1214/aos/1176347023" class="nowrap">doi:10.1214/aos/1176347023</a></td> </tr><tr> <td>[36]</td> <td class="space">Minsker, S., “Sub-Gaussian Estimators of the Mean of a Random Matrix With Heavy-Tailed Entries,”, The Annals of Statistics, 46, 2871-2903 (2018) · <a href="/1418.62235" class="nowrap">Zbl 1418.62235</a> · <a href="https://doi.org/10.1214/17-AOS1642" class="nowrap">doi:10.1214/17-AOS1642</a></td> </tr><tr> <td>[37]</td> <td class="space">Mizera, I., “On Depth and Deep Points: A Calculus,”, The Annals of Statistics, 30, 1681-1736 (2002) · <a href="/1039.62046" class="nowrap">Zbl 1039.62046</a> · <a href="https://doi.org/10.1214/aos/1043351254" class="nowrap">doi:10.1214/aos/1043351254</a></td> </tr><tr> <td>[38]</td> <td class="space">Mizera, I.; Müller, C. H., “Location-Scale Depth,”, Journal of the American Statistical Association, 99, 949-966 (2004) · <a href="/1071.62032" class="nowrap">Zbl 1071.62032</a> · <a href="https://doi.org/10.1198/016214504000001312" class="nowrap">doi:10.1198/016214504000001312</a></td> </tr><tr> <td>[39]</td> <td class="space">Nakata, B.; Takashima, T.; Ogawa, Y.; Ishikawa, T.; Hirakawa, K., “Serum CYFRA 21-1 (Cytokeratin-19 Fragments) Is a Useful Tumour Marker for Detecting Disease Relapse and Assessing Treatment Efficacy in Breast Cancer,”, British Journal of Cancer, 91, 873-878 (2004) · <a href="https://doi.org/10.1038/sj.bjc.6602074" class="nowrap">doi:10.1038/sj.bjc.6602074</a></td> </tr><tr> <td>[40]</td> <td class="space">Oh, J. H.; Yang, J. O.; Hahn, Y.; Kim, M. R.; Byun, S. S.; Jeon, Y. J.; Kim, J. M.; Song, K. S.; Noh, S. M.; Kim, S.; Yoo, H. S., “Transcriptome Analysis of Human Gastric Cancer,”, Mammalian Genome, 16, 942-954 (2005) · <a href="https://doi.org/10.1007/s00335-005-0075-2" class="nowrap">doi:10.1007/s00335-005-0075-2</a></td> </tr><tr> <td>[41]</td> <td class="space">Portnoy, S., “Asymptotic Behavior of M Estimators of p Regression Parameters When $####$ Is Large; II. Normal Approximation,”, The Annals of Statistics, 13, 1403-1417 (1985) · <a href="/0601.62026" class="nowrap">Zbl 0601.62026</a> · <a href="https://doi.org/10.1214/aos/1176349744" class="nowrap">doi:10.1214/aos/1176349744</a></td> </tr><tr> <td>[42]</td> <td class="space">Purdom, E.; Holmes, S. P., “Error Distribution for Gene Expression Data,”, Statistical Applications in Genetics and Molecular Biology, 4, 16 (2005) · <a href="/1083.62114" class="nowrap">Zbl 1083.62114</a></td> </tr><tr> <td>[43]</td> <td class="space">Ross, D. T.; Scherf, U.; Eisen, M. B.; Perou, C. M.; Rees, C.; Spellman, P.; Iyer, W.; Jeffrey, S. S.; Van de Rijn, M.; Pergamenschikov, A.; Lee, J. C. F.; Lashkari, D.; Shalon, D.; Myers, T. G.; Weinstein, J. N.; Botstein, D.; Brown, P. O., “Systematic Variation in Gene Expression Patterns in Human Cancer Cell Lines,”, Nature Genetics, 24, 227-235 (2000) · <a href="https://doi.org/10.1038/73432" class="nowrap">doi:10.1038/73432</a></td> </tr><tr> <td>[44]</td> <td class="space">Shangguan, L.; Ti, X.; Krause, U.; Hai, B.; Zhao, Y.; Yang, Z.; Liu, F., “Inhibition of TGF-β/Smad Signaling by BAMBI Blocks Differentiation of Human Mesenchymal Stem Cells to Carcinoma-Associated Fibroblasts and Abolishes Their Protumor Effects, Stem Cells, 30, 2810-2819 (2012) · <a href="https://doi.org/10.1002/stem.1251" class="nowrap">doi:10.1002/stem.1251</a></td> </tr><tr> <td>[45]</td> <td class="space">Shankavaram, U. T.; Reinhold, W. C.; Nishizuka, S.; Major, S.; Morita, D.; Chary, K. K.; Reimers, M. A.; Scherf, U.; Kahn, A.; Dolginow, D.; Cossman, J.; Kaldjian, E. P.; Scudiero, D. A.; Petricoin, E.; Liotta, L.; Lee, J. K.; Weinstein, J. N., “Transcript and Protein Expression Profiles of the NCI-60 Cancer Cell Panel: An Integromic Microarray Study,”, Molecular Cancer Therapeutics, 40, 2877-2909 (2007)</td> </tr><tr> <td>[46]</td> <td class="space">Shehata, M.; Bièche, I.; Boutros, R.; Weidenhofer, J.; Fanayan, S.; Spalding, L.; Zeps, N.; Byth, K.; Bright, R. K.; Lidereau, R.; Byrne, J. A., “Nonredundant Functions for Tumor Protein D52-Like Proteins Support Specific Targeting of TPD52,”, Clinical Cancer Research, 14, 5050-5060 (2008) · <a href="https://doi.org/10.1158/1078-0432.CCR-07-4994" class="nowrap">doi:10.1158/1078-0432.CCR-07-4994</a></td> </tr><tr> <td>[47]</td> <td class="space">Tibshirani, R., “Regression Shrinkage and Selection via the Lasso,”, Journal of the Royal Statistical Society, Series B, 58, 267-288 (1996) · <a href="/0850.62538" class="nowrap">Zbl 0850.62538</a> · <a href="https://doi.org/10.1111/j.2517-6161.1996.tb02080.x" class="nowrap">doi:10.1111/j.2517-6161.1996.tb02080.x</a></td> </tr><tr> <td>[48]</td> <td class="space">Tukey, J. W., Mathematics and the Picturing of Data, 2, 523-531 (1975) · <a href="/0347.62002" class="nowrap">Zbl 0347.62002</a></td> </tr><tr> <td>[49]</td> <td class="space">Wainwright, M. J., Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using $####$-Constrained Quadratic Programming (Lasso),”, IEEE Transactions on Information Theory, 55, 2183-2202 (2009) · <a href="/1367.62220" class="nowrap">Zbl 1367.62220</a> · <a href="https://doi.org/10.1109/TIT.2009.2016018" class="nowrap">doi:10.1109/TIT.2009.2016018</a></td> </tr><tr> <td>[50]</td> <td class="space">Wang, L., “The L_1 Penalized LAD Estimator for High Dimensional Linear Regression,”, Journal of Multivariate Analysis, 120, 135-151 (2013) · <a href="/1279.62144" class="nowrap">Zbl 1279.62144</a> · <a href="https://doi.org/10.1016/j.jmva.2013.04.001" class="nowrap">doi:10.1016/j.jmva.2013.04.001</a></td> </tr><tr> <td>[51]</td> <td class="space">Wang, L.; Peng, B.; Li, R., “A High-Dimensional Nonparametric Multivariate Test for Mean Vector,”, Journal of the American Statistical Association, 110, 1658-1669 (2015) · <a href="/1373.62280" class="nowrap">Zbl 1373.62280</a> · <a href="https://doi.org/10.1080/01621459.2014.988215" class="nowrap">doi:10.1080/01621459.2014.988215</a></td> </tr><tr> <td>[52]</td> <td class="space">Wu, Y.; Siadaty, M. S.; Berens, M. E.; Hampton, G. M.; Theodorescu, D., “Overlapping Gene Expression Profiles of Cell Migration and Tumor Invasion in Human Bladder Cancer Identify Metallothionein E1 and Nicotinamide N-Methyltransferase as Novel Regulators of Cell Migration,”, Oncogene, 27, 6679-6689 (2008) · <a href="https://doi.org/10.1038/onc.2008.264" class="nowrap">doi:10.1038/onc.2008.264</a></td> </tr><tr> <td>[53]</td> <td class="space">Yohai, V. J.; Maronna, R. A., “Asymptotic Behavior of M-Estimators for the Linear Model,”, The Annals of Statistics, 7, 258-268 (1979) · <a href="/0408.62027" class="nowrap">Zbl 0408.62027</a> · <a href="https://doi.org/10.1214/aos/1176344610" class="nowrap">doi:10.1214/aos/1176344610</a></td> </tr><tr> <td>[54]</td> <td class="space">Zheng, Q.; Peng, L.; He, X., “Globally Adaptive Quantile Regression With Ultra-High Dimensional Data,, The Annals of Statistics, 43, 2225-2258 (2015) · <a href="/1327.62424" class="nowrap">Zbl 1327.62424</a> · <a href="https://doi.org/10.1214/15-AOS1340" class="nowrap">doi:10.1214/15-AOS1340</a></td> </tr><tr> <td>[55]</td> <td class="space">Zhou, T.; Li, Y.; Yang, L.; Liu, L.; Ju, Y.; Li, C., “Silencing of ANXA3 Expression by RNA Interference Inhibits the Proliferation and Invasion of Breast Cancer Cells,”, Oncology Reports, 37, 388-398 (2017) · <a href="https://doi.org/10.3892/or.2016.5251" class="nowrap">doi:10.3892/or.2016.5251</a></td> </tr><tr> <td>[56]</td> <td class="space">Zuo, Y.; Serfling, R., “General Notions of Statistical Depth Function,”, The Annals of Statistics, 28, 461-482 (2000) · <a href="/1106.62334" class="nowrap">Zbl 1106.62334</a> · <a href="https://doi.org/10.1214/aos/1016218226" class="nowrap">doi:10.1214/aos/1016218226</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

CINXE.COM

Document Zbl 1437.62250 - zbMATH Open