Document Zbl 1503.62045

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1503.62045 - 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="/1503.62045?"> <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:1503.62045" 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/xiao.juxia" title="Author Profile">Xiao, Juxia</a>; <a href="/authors/yu.ping" title="Author Profile">Yu, Ping</a>; <a href="/authors/song.xinyuan" title="Author Profile">Song, Xinyuan</a>; <a href="/authors/zhang.zhongzhan" title="Author Profile">Zhang, Zhongzhan</a></div> <h2 class="title"> <strong>Statistical inference in the partial functional linear expectile regression model.</strong> <i>(English)</i> <a class="label nowrap" href="/1503.62045">Zbl 1503.62045</a> </h2> <div class="source"> <a href="/serials/6886" title="Journal Profile">Sci. China, Math.</a> <a href="/?q=in%3A490345" title="Articles in this Issue">65, No. 12, 2601-2630 (2022)</a>. </div> <div class="abstract">Summary: As extensions of means, expectiles embrace all the distribution information of a random variable. The expectile regression is computationally friendlier because the asymmetric least square loss function is differentiable everywhere. This regression also enables effective estimation of the expectiles of a response variable when potential explanatory variables are given. In this study, we propose the partial functional linear expectile regression model. The slope function and constant coefficients are estimated by using the functional principal component basis. The convergence rate of the slope function and the asymptotic normality of the parameter vector are established. To inspect the effect of the parametric component on the response variable, we develop Wald-type and expectile rank score tests and establish their asymptotic properties. The finite performance of the proposed estimators and test statistics are evaluated through simulation study. Results indicate that the proposed estimators are comparable to competing estimation methods and the newly proposed expectile rank score test is useful. The methodologies are illustrated by using two real data examples.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7625341">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G08" title="MSC2020">62G08</a> </td> <td class="space"> Nonparametric regression and quantile regression </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G10" title="MSC2020">62G10</a> </td> <td class="space"> Nonparametric hypothesis testing </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J05" title="MSC2020">62J05</a> </td> <td class="space"> Linear regression; mixed models </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62H25" title="MSC2020">62H25</a> </td> <td class="space"> Factor analysis and principal components; correspondence analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62R10" title="MSC2020">62R10</a> </td> <td class="space"> Functional data analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G20" title="MSC2020">62G20</a> </td> <td class="space"> Asymptotic properties of nonparametric inference </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aexpectile+regression">expectile regression</a>; <a href="/?q=ut%3Afunctional+principal+component+analysis">functional principal component analysis</a>; <a href="/?q=ut%3AWald-type+test">Wald-type test</a>; <a href="/?q=ut%3Aexpectile+rank+score+test">expectile rank score test</a>; <a href="/?q=ut%3Aheteroscedasticity">heteroscedasticity</a></div> <div class="software"> <h3>Software:</h3><a href="/software/14011">fda.usc</a>; <a href="/software/4511">fda (R)</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 1503.62045" data-ciurl="/ci/07625341" data-biburl="/bibtex/07625341.bib" data-amsurl="/amsrefs/07625341.bib" data-xmlurl="/xml/07625341.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07625341.pdf" title="Zbl 1503.62045 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/10.1007/s11425-020-1848-8" aria-label="DOI for “Statistical inference in the partial functional linear expectile regression model”" title="10.1007/s11425-020-1848-8">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abdous, B.; Remillard, B., Relating quantiles and expectiles under weighted-symmetry, Ann Inst Statist Math, 47, 371-384 (1995) · <a href="/0833.62013" class="nowrap">Zbl 0833.62013</a> · <a href="https://doi.org/10.1007/BF00773468" class="nowrap">doi:10.1007/BF00773468</a></td> </tr><tr> <td>[2]</td> <td class="space">Aneiros-Pérez, G.; Vieu, P., Semi-functional partial linear regression, Statist Probab Lett, 76, 1102-1110 (2006) · <a href="/1090.62036" class="nowrap">Zbl 1090.62036</a> · <a href="https://doi.org/10.1016/j.spl.2005.12.007" class="nowrap">doi:10.1016/j.spl.2005.12.007</a></td> </tr><tr> <td>[3]</td> <td class="space">Burdejová, P.; Härdle, W., Dynamic semi-parametric factor model for functional expectiles, Comput Statist, 34, 489-502 (2019) · <a href="/1417.62163" class="nowrap">Zbl 1417.62163</a> · <a href="https://doi.org/10.1007/s00180-019-00883-1" class="nowrap">doi:10.1007/s00180-019-00883-1</a></td> </tr><tr> <td>[4]</td> <td class="space">Cai, T.; Hall, P., Prediction in functional linear regression, Ann Statist, 34, 2159-2179 (2006) · <a href="/1106.62036" class="nowrap">Zbl 1106.62036</a> · <a href="https://doi.org/10.1214/009053606000000830" class="nowrap">doi:10.1214/009053606000000830</a></td> </tr><tr> <td>[5]</td> <td class="space">Cui, X.; Lin, H.; Lian, H., Partially functional linear regression in reproducing kernel Hilbert spaces, Comput Statist Data Anal, 150, 106978 (2020) · <a href="/1510.62171" class="nowrap">Zbl 1510.62171</a> · <a href="https://doi.org/10.1016/j.csda.2020.106978" class="nowrap">doi:10.1016/j.csda.2020.106978</a></td> </tr><tr> <td>[6]</td> <td class="space">Daouia, A.; Girard, S.; Stupfler, G., Estimation of tail risk based on extreme expectiles, J R Stat Soc Ser B Stat Methodol, 80, 263-292 (2018) · <a href="/1383.62235" class="nowrap">Zbl 1383.62235</a> · <a href="https://doi.org/10.1111/rssb.12254" class="nowrap">doi:10.1111/rssb.12254</a></td> </tr><tr> <td>[7]</td> <td class="space">Daouia, A.; Girard, S.; Stupfler, G., Extremiles: A new perspective on asymmetric least squares, J Amer Statist Assoc, 114, 1366-1381 (2019) · <a href="/1428.62198" class="nowrap">Zbl 1428.62198</a> · <a href="https://doi.org/10.1080/01621459.2018.1498348" class="nowrap">doi:10.1080/01621459.2018.1498348</a></td> </tr><tr> <td>[8]</td> <td class="space">Febrero-bande, M.; Fuente, M., Statistical computing in functional data analysis: The R package fda.usc, J Statist Softw, 51, 1-28 (2012) · <a href="https://doi.org/10.18637/jss.v051.i04" class="nowrap">doi:10.18637/jss.v051.i04</a></td> </tr><tr> <td>[9]</td> <td class="space">Gu, Y.; Zou, H., High-dimensional generalizations of asymmetric least squares regression and their applications, Ann Statist, 44, 2661-2694 (2016) · <a href="/1364.62185" class="nowrap">Zbl 1364.62185</a> · <a href="https://doi.org/10.1214/15-AOS1431" class="nowrap">doi:10.1214/15-AOS1431</a></td> </tr><tr> <td>[10]</td> <td class="space">Guo, M.; Zhou, L.; Huang, J., Functional data analysis of generalized regression quantiles, Stat Comput, 25, 189-202 (2015) · <a href="/1331.62031" class="nowrap">Zbl 1331.62031</a> · <a href="https://doi.org/10.1007/s11222-013-9425-1" class="nowrap">doi:10.1007/s11222-013-9425-1</a></td> </tr><tr> <td>[11]</td> <td class="space">Gutenbrunner, C.; Jurečková, J.; Koenker, R., Tests of linear hypotheses based on regression rank scores, J Nonparametr Stat, 2, 307-331 (1993) · <a href="/1360.62216" class="nowrap">Zbl 1360.62216</a> · <a href="https://doi.org/10.1080/10485259308832561" class="nowrap">doi:10.1080/10485259308832561</a></td> </tr><tr> <td>[12]</td> <td class="space">Hall, P.; Hooker, G., Truncated linear models for functional data, J R Stat Soc Ser B Stat Methodol, 78, 637-653 (2016) · <a href="/1414.62181" class="nowrap">Zbl 1414.62181</a> · <a href="https://doi.org/10.1111/rssb.12125" class="nowrap">doi:10.1111/rssb.12125</a></td> </tr><tr> <td>[13]</td> <td class="space">Hall, P.; Horowitz, J., Methodology and convergence rates for functional linear regression, Ann Statist, 35, 70-91 (2007) · <a href="/1114.62048" class="nowrap">Zbl 1114.62048</a> · <a href="https://doi.org/10.1214/009053606000000957" class="nowrap">doi:10.1214/009053606000000957</a></td> </tr><tr> <td>[14]</td> <td class="space">Holzmann, H.; Klar, B., Expectile asymptotics, Electron J Stat, 10, 2355-2371 (2016) · <a href="/1397.62075" class="nowrap">Zbl 1397.62075</a> · <a href="https://doi.org/10.1214/16-EJS1173" class="nowrap">doi:10.1214/16-EJS1173</a></td> </tr><tr> <td>[15]</td> <td class="space">Imaizumi, M.; Kato, K., PCA-based estimation for functional linear regression with functional responses, J Multivariate Anal, 163, 15-36 (2018) · <a href="/1499.62137" class="nowrap">Zbl 1499.62137</a> · <a href="https://doi.org/10.1016/j.jmva.2017.10.001" class="nowrap">doi:10.1016/j.jmva.2017.10.001</a></td> </tr><tr> <td>[16]</td> <td class="space">Kato, K., Estimation in functional linear quantile regression, Ann Statist, 40, 3108-3136 (2012) · <a href="/1296.62104" class="nowrap">Zbl 1296.62104</a> · <a href="https://doi.org/10.1214/12-AOS1066" class="nowrap">doi:10.1214/12-AOS1066</a></td> </tr><tr> <td>[17]</td> <td class="space">Kong, D.; Xue, K.; Yao, F., Partially functional linear regression in high dimensions, Biometrika, 103, 147-159 (2016) · <a href="/1452.62500" class="nowrap">Zbl 1452.62500</a> · <a href="https://doi.org/10.1093/biomet/asv062" class="nowrap">doi:10.1093/biomet/asv062</a></td> </tr><tr> <td>[18]</td> <td class="space">Krätschmer, V.; Zähle, H., Statistical inference for expectile-based risk measures, Scand J Stat, 44, 425-454 (2017) · <a href="/1422.62241" class="nowrap">Zbl 1422.62241</a></td> </tr><tr> <td>[19]</td> <td class="space">Liao, L.; Park, C.; Choi, H., Penalized expectile regression an alternative to penalized quantile regression, Ann Inst Statist Math, 71, 409-438 (2019) · <a href="/1417.62082" class="nowrap">Zbl 1417.62082</a> · <a href="https://doi.org/10.1007/s10463-018-0645-1" class="nowrap">doi:10.1007/s10463-018-0645-1</a></td> </tr><tr> <td>[20]</td> <td class="space">Lu, Y.; Du, J.; Sun, Z., Functional partially linear quantile regression model, Metrika, 77, 317-332 (2014) · <a href="/1304.62058" class="nowrap">Zbl 1304.62058</a> · <a href="https://doi.org/10.1007/s00184-013-0439-7" class="nowrap">doi:10.1007/s00184-013-0439-7</a></td> </tr><tr> <td>[21]</td> <td class="space">Ma, H. Q.; Bai, Y.; Zhu, Z. Y., Dynamic single-index model for functional data, Sci China Math, 59, 2561-2584 (2016) · <a href="/1360.62185" class="nowrap">Zbl 1360.62185</a> · <a href="https://doi.org/10.1007/s11425-016-0051-3" class="nowrap">doi:10.1007/s11425-016-0051-3</a></td> </tr><tr> <td>[22]</td> <td class="space">Newey, W.; Powell, J., Asymmetric least squares estimation and testing, Econometrica, 55, 819-847 (1987) · <a href="/0625.62047" class="nowrap">Zbl 0625.62047</a> · <a href="https://doi.org/10.2307/1911031" class="nowrap">doi:10.2307/1911031</a></td> </tr><tr> <td>[23]</td> <td class="space">Osborne, B.; Fearn, T.; Miller, A., Application of near infrared reflectance spectroscopy to the compositional analysis of biscuits and biscuit doughs, J Sci Food Agriculture, 35, 99-105 (1984) · <a href="https://doi.org/10.1002/jsfa.2740350116" class="nowrap">doi:10.1002/jsfa.2740350116</a></td> </tr><tr> <td>[24]</td> <td class="space">Ramsay, J.; Silverman, B., Functional Data Analysis (2005), New York: Springer, New York · <a href="/1079.62006" class="nowrap">Zbl 1079.62006</a> · <a href="https://doi.org/10.1007/b98888" class="nowrap">doi:10.1007/b98888</a></td> </tr><tr> <td>[25]</td> <td class="space">Shin, H., Partial functional linear regression, J Statist Plann Inference, 139, 3405-3418 (2009) · <a href="/1168.62358" class="nowrap">Zbl 1168.62358</a> · <a href="https://doi.org/10.1016/j.jspi.2009.03.001" class="nowrap">doi:10.1016/j.jspi.2009.03.001</a></td> </tr><tr> <td>[26]</td> <td class="space">Shin, H.; Lee, M., On prediction rate in partial functional linear regression, J Multivariate Anal, 103, 93-106 (2012) · <a href="/1231.62072" class="nowrap">Zbl 1231.62072</a> · <a href="https://doi.org/10.1016/j.jmva.2011.06.011" class="nowrap">doi:10.1016/j.jmva.2011.06.011</a></td> </tr><tr> <td>[27]</td> <td class="space">Sobotka, F.; Kneib, T., Geoadditive expectile regression, Comput Statist Data Anal, 56, 755-767 (2012) · <a href="/1241.62058" class="nowrap">Zbl 1241.62058</a> · <a href="https://doi.org/10.1016/j.csda.2010.11.015" class="nowrap">doi:10.1016/j.csda.2010.11.015</a></td> </tr><tr> <td>[28]</td> <td class="space">Tang, Q. G.; Cheng, L. S., Partial functional linear quantile regression, Sci China Math, 57, 2589-2608 (2014) · <a href="/1308.62086" class="nowrap">Zbl 1308.62086</a> · <a href="https://doi.org/10.1007/s11425-014-4819-x" class="nowrap">doi:10.1007/s11425-014-4819-x</a></td> </tr><tr> <td>[29]</td> <td class="space">Tran, N.; Burdejová, P.; Osipenko, M., Principal component analysis in an asymmetric norm, J Multivariate Anal, 171, 1-21 (2019) · <a href="/1418.62249" class="nowrap">Zbl 1418.62249</a> · <a href="https://doi.org/10.1016/j.jmva.2018.10.004" class="nowrap">doi:10.1016/j.jmva.2018.10.004</a></td> </tr><tr> <td>[30]</td> <td class="space">Wang, H.; Fygenson, M., Inference for censored quantile regression models in longitudinal studies, Ann Statist, 37, 756-781 (2009) · <a href="/1162.62035" class="nowrap">Zbl 1162.62035</a> · <a href="https://doi.org/10.1214/07-AOS564" class="nowrap">doi:10.1214/07-AOS564</a></td> </tr><tr> <td>[31]</td> <td class="space">Wang, H.; Zhu, Z.; Zhou, J., Quantile regression in partially linear varying coefficient models, Ann Statist, 37, 3841-3866 (2009) · <a href="/1191.62077" class="nowrap">Zbl 1191.62077</a> · <a href="https://doi.org/10.1214/09-AOS695" class="nowrap">doi:10.1214/09-AOS695</a></td> </tr><tr> <td>[32]</td> <td class="space">Xing, J.; Qian, X., Bayesian expectile regression with asymmetric normal distribution, Comm Statist Theory Methods, 46, 4545-4555 (2017) · <a href="/1422.62112" class="nowrap">Zbl 1422.62112</a> · <a href="https://doi.org/10.1080/03610926.2015.1088030" class="nowrap">doi:10.1080/03610926.2015.1088030</a></td> </tr><tr> <td>[33]</td> <td class="space">Yao, F.; Müller, H.; Wang, J., Functional linear regression analysis for longitudinal data, Ann Statist, 33, 2873-2903 (2005) · <a href="/1084.62096" class="nowrap">Zbl 1084.62096</a> · <a href="https://doi.org/10.1214/009053605000000660" class="nowrap">doi:10.1214/009053605000000660</a></td> </tr><tr> <td>[34]</td> <td class="space">Yu, P.; Zhang, Z.; Du, J., A test of linearity in partial functional linear regression, Metrika, 79, 953-969 (2016) · <a href="/1358.62045" class="nowrap">Zbl 1358.62045</a> · <a href="https://doi.org/10.1007/s00184-016-0584-x" class="nowrap">doi:10.1007/s00184-016-0584-x</a></td> </tr><tr> <td>[35]</td> <td class="space">Yu, P.; Zhu, Z.; Zhang, Z., Robust exponential squared loss-based estimation in semi-functional linear regression models, Comput Statist, 34, 503-525 (2019) · <a href="/1417.62103" class="nowrap">Zbl 1417.62103</a> · <a href="https://doi.org/10.1007/s00180-018-0810-2" class="nowrap">doi:10.1007/s00180-018-0810-2</a></td> </tr><tr> <td>[36]</td> <td class="space">Yuan, M.; Cai, T., A reproducing kernel Hilbert space approach to functional linear regression, Ann Statist, 38, 3412-3444 (2010) · <a href="/1204.62074" class="nowrap">Zbl 1204.62074</a> · <a href="https://doi.org/10.1214/09-AOS772" class="nowrap">doi:10.1214/09-AOS772</a></td> </tr><tr> <td>[37]</td> <td class="space">Zhang, F.; Li, Q., A continuous threshold expectile model, Comput Statist Data Anal, 116, 49-66 (2017) · <a href="/1466.62225" class="nowrap">Zbl 1466.62225</a> · <a href="https://doi.org/10.1016/j.csda.2017.07.005" class="nowrap">doi:10.1016/j.csda.2017.07.005</a></td> </tr><tr> <td>[38]</td> <td class="space">Zhou, J.; Chen, Z.; Peng, Q., Polynomial spline estimation for partial functional linear regression models, Comput Statist, 31, 1107-1129 (2016) · <a href="/1347.65036" class="nowrap">Zbl 1347.65036</a> · <a href="https://doi.org/10.1007/s00180-015-0636-0" class="nowrap">doi:10.1007/s00180-015-0636-0</a></td> </tr><tr> <td>[39]</td> <td class="space">Zhu, H.; Zhang, R.; Yu, Z., Estimation and testing for partially functional linear errors-in-variables models, J Multivariate Anal, 170, 296-314 (2019) · <a href="/1420.62186" class="nowrap">Zbl 1420.62186</a> · <a href="https://doi.org/10.1016/j.jmva.2018.11.005" class="nowrap">doi:10.1016/j.jmva.2018.11.005</a></td> </tr><tr> <td>[40]</td> <td class="space">Zhu, S.; Zhao, P., Tests for the linear hypothesis in semi-functional partial linear regression models, Metrika, 82, 125-148 (2019) · <a href="/1419.62097" class="nowrap">Zbl 1419.62097</a> · <a href="https://doi.org/10.1007/s00184-018-0680-1" class="nowrap">doi:10.1007/s00184-018-0680-1</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 1503.62045 - zbMATH Open