Document Zbl 1480.62102

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1480.62102 - 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="/?q=ra%3Axiao.juxia+ai%3Atolosana-delgado.raimon"> <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="ra:xiao.juxia ai:tolosana-delgado.raimon" 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/ouimet.frederic" title="Author Profile">Ouimet, Frédéric</a>; <a href="/authors/tolosana-delgado.raimon" title="Author Profile">Tolosana-Delgado, Raimon</a></div> <h2 class="title"> <strong>Asymptotic properties of Dirichlet kernel density estimators.</strong> <i>(English)</i> <a class="label nowrap" href="/1480.62102">Zbl 1480.62102</a> </h2> <div class="source"> <a href="/serials/510" title="Journal Profile">J. Multivariate Anal.</a> <a href="/?q=in%3A473024" title="Articles in this Issue">187, Article ID 104832, 25 p. (2022)</a>. </div> <div class="abstract">Summary: We study theoretically, for the first time, the Dirichlet kernel estimator introduced by <span class="zbmathjax-textit">J. Aitchison</span> and <span class="zbmathjax-textit">I. J. Lauder</span> [J. R. Stat. Soc., Ser. C 34, 129–137 (1985; <a href="/0585.62069">Zbl 0585.62069</a>)] for the estimation of multivariate densities supported on the $d$-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. The Dirichlet kernel estimator, which generalizes the (non-modified) unidimensional Beta kernel estimator from [<span class="zbmathjax-textit">S. X. Chen</span>, Comput. Stat. Data Anal. 31, No. 2, 131–145 (1999; <a href="/0935.62042">Zbl 0935.62042</a>)], is free of boundary bias and non-negative everywhere on the simplex. We show that it achieves the optimal convergence rate $\mathcal{O} ( n^{- 4 / ( d + 4 )} )$ for the mean squared error and the mean integrated squared error, we prove its asymptotic normality and uniform strong consistency, and we also find an asymptotic expression for the mean integrated absolute error. To illustrate the Dirichlet kernel method and its favorable boundary properties, we present a case study on minerals processing.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A7439797">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A7439797">Cited in <strong>4</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62H12" title="MSC2020">62H12</a> </td> <td class="space"> Estimation in multivariate analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G07" title="MSC2020">62G07</a> </td> <td class="space"> Density estimation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62G05" title="MSC2020">62G05</a> </td> <td class="space"> Nonparametric estimation </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%3ADirichlet+kernel">Dirichlet kernel</a>; <a href="/?q=ut%3Abeta+kernel">beta kernel</a>; <a href="/?q=ut%3Aasymmetric+kernel">asymmetric kernel</a>; <a href="/?q=ut%3Adensity+estimation">density estimation</a>; <a href="/?q=ut%3Asimplex">simplex</a>; <a href="/?q=ut%3Aboundary+bias">boundary bias</a>; <a href="/?q=ut%3Avariance">variance</a>; <a href="/?q=ut%3Amean+squared+error">mean squared error</a>; <a href="/?q=ut%3Amean+integrated+absolute+error">mean integrated absolute error</a>; <a href="/?q=ut%3Aasymptotic+normality">asymptotic normality</a>; <a href="/?q=ut%3Astrong+consistency">strong consistency</a>; <a href="/?q=ut%3Amultivariate+associated+kernel">multivariate associated kernel</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0585.62069">Zbl 0585.62069</a>; <a href="/0935.62042">Zbl 0935.62042</a></div> <div class="software"> <h3>Software:</h3><a href="/software/40824">Disake</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 1480.62102" data-ciurl="/ci/07439797" data-biburl="/bibtex/07439797.bib" data-amsurl="/amsrefs/07439797.bib" data-xmlurl="/xml/07439797.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07439797.pdf" title="Zbl 1480.62102 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/10.1016/j.jmva.2021.104832" aria-label="DOI for “Asymptotic properties of Dirichlet kernel density estimators”" title="10.1016/j.jmva.2021.104832">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2002.06956"title="Note: arXiv document may differ from published version">arXiv</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.; Kokonendji, C. C., Consistency and asymptotic normality for discrete associated-kernel estimator, Afr. Diaspora J. Math. (N.S.), 8, 2, 63-70 (2009), MR2511101 · <a href="/1239.62032" class="nowrap">Zbl 1239.62032</a></td> </tr><tr> <td>[2]</td> <td class="space">Abramowitz, M.; Stegun, I. A., (Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55 (1964), For sale by the Superintendent of Documents, U.S. Government Printing Office: For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C.), xiv+1046, MR0167642 · <a href="/0171.38503" class="nowrap">Zbl 0171.38503</a></td> </tr><tr> <td>[3]</td> <td class="space">Aitchison, J.; Lauder, I. J., Kernel density estimation for compositional data, J. Roy. Statist. Soc. Ser. C, 34, 2, 129-137 (1985) · <a href="/0585.62069" class="nowrap">Zbl 0585.62069</a></td> </tr><tr> <td>[4]</td> <td class="space">Babu, G. J.; Canty, A. J.; Chaubey, Y. P., Application of Bernstein polynomials for smooth estimation of a distribution and density function, J. Statist. Plann. Inference, 105, 2, 377-392 (2002), MR1910059 · <a href="/0992.62038" class="nowrap">Zbl 0992.62038</a></td> </tr><tr> <td>[5]</td> <td class="space">Babu, G. J.; Chaubey, Y. P., Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors, Statist. Probab. Lett., 76, 9, 959-969 (2006), MR2270097 · <a href="/1089.62034" class="nowrap">Zbl 1089.62034</a></td> </tr><tr> <td>[6]</td> <td class="space">Batır, N., Bounds for the gamma function, Results Math., 72, 1-2, 865-874 (2017), MR3684463 · <a href="/1371.33003" class="nowrap">Zbl 1371.33003</a></td> </tr><tr> <td>[7]</td> <td class="space">Belaid, N.; Adjabi, S.; Kokonendji, C. C.; Zougab, N., Bayesian local bandwidth selector in multivariate associated kernel estimator for joint probability mass functions, J. Stat. Comput. Simul., 86, 18, 3667-3681 (2016), MR3547951 · <a href="/07184822" class="nowrap">Zbl 07184822</a></td> </tr><tr> <td>[8]</td> <td class="space">Belaid, N.; Adjabi, S.; Zougab, N.; Kokonendji, C. C., Bayesian bandwidth selection in discrete multivariate associated kernel estimators for probability mass functions, J. Korean Statist. Soc., 45, 4, 557-567 (2016), MR3566162 · <a href="/1348.62115" class="nowrap">Zbl 1348.62115</a></td> </tr><tr> <td>[9]</td> <td class="space">Belalia, M., On the asymptotic properties of the Bernstein estimator of the multivariate distribution function, Statist. Probab. Lett., 110, 249-256 (2016), MR3474765 · <a href="/1381.62093" class="nowrap">Zbl 1381.62093</a></td> </tr><tr> <td>[10]</td> <td class="space">Bertin, K.; El Kolei, S.; Klutchnikoff, N., Adaptive density estimation on bounded domains, Ann. Inst. Henri Poincaré Probab. Stat., 55, 4, 1916-1947 (2019), MR4029144 · <a href="/1436.62118" class="nowrap">Zbl 1436.62118</a></td> </tr><tr> <td>[11]</td> <td class="space">Bertin, K.; Klutchnikoff, N., Minimax properties of beta kernel estimators, J. Statist. Plann. Inference, 141, 7, 2287-2297 (2011), MR2775207 · <a href="/1214.62038" class="nowrap">Zbl 1214.62038</a></td> </tr><tr> <td>[12]</td> <td class="space">Bertin, K.; Klutchnikoff, N., Adaptive estimation of a density function using beta kernels, ESAIM Probab. Stat., 18, 400-417 (2014), MR3333996 · <a href="/1305.62140" class="nowrap">Zbl 1305.62140</a></td> </tr><tr> <td>[13]</td> <td class="space">Bertin, K.; Klutchnikoff, N.; Léon, J. R.; Prieur, C., Adaptive density estimation on bounded domains under mixing conditions, Electron. J. Stat., 14, 1, 2198-2237 (2020), MR4097810 · <a href="/1442.62076" class="nowrap">Zbl 1442.62076</a></td> </tr><tr> <td>[14]</td> <td class="space">Bouezmarni, T.; van Bellegem, S., Nonparametric Beta Kernel Estimator for Long Memory Time SeriesCORE Discussion Paper, 1-20 (2011), [URL] https://ideas.repec.org/p/cor/louvco/2011004.html</td> </tr><tr> <td>[15]</td> <td class="space">Bouezmarni, T.; El Ghouch, A.; Mesfioui, M., Gamma kernel estimators for density and hazard rate of right-censored data, J. Probab. Stat., Article 937574 pp. (2011), 16 pp, MR2801351 · <a href="/1229.62041" class="nowrap">Zbl 1229.62041</a></td> </tr><tr> <td>[16]</td> <td class="space">Bouezmarni, T.; Rolin, J.-M., Consistency of the beta kernel density function estimator, Canad. J. Statist., 31, 1, 89-98 (2003), MR1985506 · <a href="/1039.62030" class="nowrap">Zbl 1039.62030</a></td> </tr><tr> <td>[17]</td> <td class="space">Bouezmarni, T.; Rolin, J.-M., Bernstein estimator for unbounded density function, J. Nonparametr. Stat., 19, 3, 145-161 (2007), MR2351744 · <a href="/1122.62018" class="nowrap">Zbl 1122.62018</a></td> </tr><tr> <td>[18]</td> <td class="space">Bouezmarni, T.; Rombouts, J. V.K., Density and hazard rate estimation for censored and $\alpha $-mixing data using gamma kernels, J. Nonparametr. Stat., 20, 7, 627-643 (2008), MR2454617 · <a href="/1147.62029" class="nowrap">Zbl 1147.62029</a></td> </tr><tr> <td>[19]</td> <td class="space">Bouezmarni, T.; Rombouts, J. V.K., Semiparametric multivariate density estimation for positive data using copulas, Comput. Statist. Data Anal., 53, 6, 2040-2054 (2009), MR2665093 · <a href="/1453.62052" class="nowrap">Zbl 1453.62052</a></td> </tr><tr> <td>[20]</td> <td class="space">Bouezmarni, T.; Rombouts, J. V.K., Nonparametric density estimation for multivariate bounded data, J. Statist. Plann. Inference, 140, 1, 139-152 (2010), MR2568128 · <a href="/1178.62026" class="nowrap">Zbl 1178.62026</a></td> </tr><tr> <td>[21]</td> <td class="space">Bouezmarni, T.; Rombouts, J. V.K., Nonparametric density estimation for positive time series, Comput. Statist. Data Anal., 54, 2, 245-261 (2010), MR2756423 · <a href="/1464.62033" class="nowrap">Zbl 1464.62033</a></td> </tr><tr> <td>[22]</td> <td class="space">Bouezmarni, T.; Scaillet, O., Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data, Econom. Theory, 21, 2, 390-412 (2005), MR2179543 · <a href="/1062.62058" class="nowrap">Zbl 1062.62058</a></td> </tr><tr> <td>[23]</td> <td class="space">Bouezmarni, T.; Van Bellegem, S.; Rabhi, Y., Nonparametric beta kernel estimator for long and short memory time series, Canad. J. Statist., 48, 3, 582-595 (2020), MR4148613 · <a href="/1492.62141" class="nowrap">Zbl 1492.62141</a></td> </tr><tr> <td>[24]</td> <td class="space">Brown, B. M.; Chen, S. X., Beta-Bernstein smoothing for regression curves with compact support, Scand. J. Stat., 26, 1, 47-59 (1999), MR1685301 · <a href="/0921.62048" class="nowrap">Zbl 0921.62048</a></td> </tr><tr> <td>[25]</td> <td class="space">Chacón, J. E.; Mateu-Figueras, G.; Martín-Fernández, J. A., Gaussian kernels for density estimation with compositional data, Comput. Geosci., 37, 702-711 (2011)</td> </tr><tr> <td>[26]</td> <td class="space">Charpentier, A., Dependence Structures and Limiting Results, with Applications in Finance and Insurance, 296 (2006), Katholieke Universiteit Leuven, [URL] https://tel.archives-ouvertes.fr/file/index/docid/82892/filename/thesis.pdf</td> </tr><tr> <td>[27]</td> <td class="space">Charpentier, A.; Fermanian, J.-D.; Scaillet, O., The estimation of copulas: theory and practice, (Rank, J., Copulas: From Theory to Application in Finance (2007), Risk Books: Risk Books London), 35-64, [URL] https://archive-ouverte.unige.ch/unige:41917</td> </tr><tr> <td>[28]</td> <td class="space">Charpentier, A.; Flachaire, E., Log-transform kernel density estimation of income distribution, Actual. Économ. Rev. Anal. Économ., 91, 1-2, 141-159 (2015)</td> </tr><tr> <td>[29]</td> <td class="space">Charpentier, A.; Oulidi, A., Beta kernel quantile estimators of heavy-tailed loss distributions, Stat. Comput., 20, 1, 35-55 (2010), MR2578075</td> </tr><tr> <td>[30]</td> <td class="space">Chaubey, Y. P.; Dewan, I.; Li, J., An asymmetric kernel estimator of density function for stationary associated sequences, Comm. Statist. Simulation Comput., 41, 4, 554-572 (2012), MR2869005 · <a href="/1294.62075" class="nowrap">Zbl 1294.62075</a></td> </tr><tr> <td>[31]</td> <td class="space">Chaubey, Y. P.; Li, J., Asymmetric kernel density estimator for length biased data, (Contemporary Topics in Mathematics and Statistics with Applications, Vol. 1 (2013), Asian Books Private Ltd), 28, URL https://spectrum.library.concordia.ca/973586/1/Chaubey_IMBIC_monograph.pdf</td> </tr><tr> <td>[32]</td> <td class="space">Chaubey, Y. P.; Li, J.; Sen, A.; Sen, P. K., A new smooth density estimator for non-negative random variables, J. Indian Statist. Assoc., 50, 1-2, 83-104 (2012), MR2975812 · <a href="/1462.62225" class="nowrap">Zbl 1462.62225</a></td> </tr><tr> <td>[33]</td> <td class="space">Chekkal, S.; Lagha, K.; Zougab, N., Generalized Birnbaum-Saunders kernel for hazard rate function estimation, Comm. Statist. Simulation Comput., 1-16 (2021)</td> </tr><tr> <td>[34]</td> <td class="space">Chen, S. X., Beta kernel estimators for density functions, Comput. Statist. Data Anal., 31, 2, 131-145 (1999), MR1718494 · <a href="/0935.62042" class="nowrap">Zbl 0935.62042</a></td> </tr><tr> <td>[35]</td> <td class="space">Chen, S. X., Beta kernel smoothers for regression curves, Statist. Sinica, 10, 1, 73-91 (2000), MR1742101 · <a href="/0970.62018" class="nowrap">Zbl 0970.62018</a></td> </tr><tr> <td>[36]</td> <td class="space">Chen, S. X., Probability density function estimation using gamma kernels, Ann. Inst. Statist. Math., 52, 3, 471-480 (2000), MR1794247 · <a href="/0960.62038" class="nowrap">Zbl 0960.62038</a></td> </tr><tr> <td>[37]</td> <td class="space">Chen, S. X., Local linear smoothers using asymmetric kernels, Ann. Inst. Statist. Math., 54, 2, 312-323 (2002), MR1910175 · <a href="/1012.62032" class="nowrap">Zbl 1012.62032</a></td> </tr><tr> <td>[38]</td> <td class="space">Comte, F.; Genon-Catalot, V., Convolution power kernels for density estimation, J. Statist. Plann. Inference, 142, 7, 1698-1715 (2012), MR2903382 · <a href="/1238.62041" class="nowrap">Zbl 1238.62041</a></td> </tr><tr> <td>[39]</td> <td class="space">Devroye, L.; Györfi, L., (Nonparametric Density Estimation: The $L_1$ View. Nonparametric Density Estimation: The $L_1$ View, Wiley Series in Probability and Mathematical Statistics (1985), John Wiley & Sons, Inc., New York), xi+356, MR780746 · <a href="/0546.62015" class="nowrap">Zbl 0546.62015</a></td> </tr><tr> <td>[40]</td> <td class="space">Devroye, L.; Penrod, C. S., Distribution-free lower bounds in density estimation, Ann. Statist., 12, 4, 1250-1262 (1984), MR760686 · <a href="/0551.62024" class="nowrap">Zbl 0551.62024</a></td> </tr><tr> <td>[41]</td> <td class="space">Devroye, L.; Penrod, C. S., The strong uniform convergence of multivariate variable kernel estimates, Canad. J. Statist., 14, 3, 211-219 (1986), MR859633 · <a href="/0615.62044" class="nowrap">Zbl 0615.62044</a></td> </tr><tr> <td>[42]</td> <td class="space">Dobrovidov, A. V.; Markovich, L. A., Data-driven bandwidth choice for gamma kernel estimates of density derivatives on the positive semi-axis, IFAC Proc. Vol., 46, 11, 500-505 (2013)</td> </tr><tr> <td>[43]</td> <td class="space">Dobrovidov, A. V.; Markovich, L. A., Nonparametric gamma kernel estimators of density derivatives on positive semi-axis, IFAC Proc. Vol., 46, 9, 910-915 (2013)</td> </tr><tr> <td>[44]</td> <td class="space">Erçelik, E.; Nadar, M., A new kernel estimator based on scaled inverse chi-squared density function, Am. J. Math. Manag. Sci., 1-14 (2020)</td> </tr><tr> <td>[45]</td> <td class="space">Erçelik, E.; Nadar, M., Nonparametric density estimation based on beta prime kernel, Comm. Statist. Theory Methods, 49, 2, 325-342 (2020), MR4037101 · <a href="/07549036" class="nowrap">Zbl 07549036</a></td> </tr><tr> <td>[46]</td> <td class="space">Fan, J.; Gijbels, I., Variable bandwidth and local linear regression smoothers, Ann. Statist., 20, 4, 2008-2036 (1992), MR1193323 · <a href="/0765.62040" class="nowrap">Zbl 0765.62040</a></td> </tr><tr> <td>[47]</td> <td class="space">Fauzi, R. R.; Maesono, Y., New type of gamma kernel density estimator, J. Korean Statist. Soc., 49, 3, 882-900 (2020), MR4147689 · <a href="/1484.62042" class="nowrap">Zbl 1484.62042</a></td> </tr><tr> <td>[48]</td> <td class="space">Fé, E., Estimation and inference in regression discontinuity designs with asymmetric kernels, J. Appl. Stat., 41, 11, 2406-2417 (2014), MR3256394 · <a href="/1514.62559" class="nowrap">Zbl 1514.62559</a></td> </tr><tr> <td>[49]</td> <td class="space">Fernandes, M.; Grammig, J., Nonparametric specification tests for conditional duration models, J. Econometrics, 127, 1, 35-68 (2005), MR2137490 · <a href="/1337.62216" class="nowrap">Zbl 1337.62216</a></td> </tr><tr> <td>[50]</td> <td class="space">Fernandes, M.; Mendes, E. F.; Scaillet, O., Testing for symmetry and conditional symmetry using asymmetric kernels, Ann. Inst. Statist. Math., 67, 4, 649-671 (2015), MR3357933 · <a href="/1440.62147" class="nowrap">Zbl 1440.62147</a></td> </tr><tr> <td>[51]</td> <td class="space">Fernandes, M.; Monteiro, P. K., Central limit theorem for asymmetric kernel functionals, Ann. Inst. Statist. Math., 57, 3, 425-442 (2005), MR2206532 · <a href="/1095.62041" class="nowrap">Zbl 1095.62041</a></td> </tr><tr> <td>[52]</td> <td class="space">Funke, B.; Hirukawa, M., Nonparametric estimation and testing on discontinuity of positive supported densities: a kernel truncation approach, Econom. Stat., 9, 156-170 (2019), MR3907679</td> </tr><tr> <td>[53]</td> <td class="space">Funke, B.; Hirukawa, M., Bias correction for local linear regression estimation using asymmetric kernels via the skewing method, Econom. Stat., 20, 109-130 (2021), MR4302589</td> </tr><tr> <td>[54]</td> <td class="space">Funke, B.; Kawka, R., Nonparametric density estimation for multivariate bounded data using two non-negative multiplicative bias correction methods, Comput. Statist. Data Anal., 92, 148-162 (2015), MR3384258 · <a href="/1468.62057" class="nowrap">Zbl 1468.62057</a></td> </tr><tr> <td>[55]</td> <td class="space">Gasser, T.; Müller, H.-G., Kernel estimation of regression functions, (Smoothing Techniques for Curve Estimation (1979), Springer Berlin Heidelberg), 23-68 · <a href="/0418.62033" class="nowrap">Zbl 0418.62033</a></td> </tr><tr> <td>[56]</td> <td class="space">Gasser, T.; Müller, H.-G.; Mammitzsch, V., Kernels for nonparametric curve estimation, J. R. Stat. Soc. Ser. B Stat. Methodol., 47, 2, 238-252 (1985), MR816088 · <a href="/0574.62042" class="nowrap">Zbl 0574.62042</a></td> </tr><tr> <td>[57]</td> <td class="space">Gawronski, W., Strong laws for density estimators of Bernstein type, Period. Math. Hungar., 16, 1, 23-43 (1985), MR0791719 · <a href="/0531.62040" class="nowrap">Zbl 0531.62040</a></td> </tr><tr> <td>[58]</td> <td class="space">Gawronski, W.; Stadtmüller, U., On density estimation by means of Poisson’s distribution, Scand. J. Stat., 7, 2, 90-94 (1980), MR0574548 · <a href="/0433.62026" class="nowrap">Zbl 0433.62026</a></td> </tr><tr> <td>[59]</td> <td class="space">Gawronski, W.; Stadtmüller, U., Smoothing histograms by means of lattice and continuous distributions, Metrika, 28, 3, 155-164 (1981), MR0638651 · <a href="/0473.62034" class="nowrap">Zbl 0473.62034</a></td> </tr><tr> <td>[60]</td> <td class="space">Gawronski, W.; Stadtmüller, U., Linear combinations of iterated generalized Bernstein functions with an application to density estimation, Acta Sci. Math., 47, 1-2, 205-221 (1984), MR0755576 · <a href="/0565.41022" class="nowrap">Zbl 0565.41022</a></td> </tr><tr> <td>[61]</td> <td class="space">Geenens, G., Mellin-meijer-kernel density estimation on $\mathbb{R}^+$, Ann. Inst. Stat. Math., 25 (2020)</td> </tr><tr> <td>[62]</td> <td class="space">Gospodinov, N.; Hirukawa, M., Nonparametric estimation of scalar diffusion models of interest rates using asymmetric kernels, J. Empir. Finance, 19, 4, 595-609 (2012)</td> </tr><tr> <td>[63]</td> <td class="space">Gouriéroux, C.; Monfort, A., (Non) Consistency of the Beta Kernel Estimator for Recovery Rate DistributionCREST Discussion Paper, 1-27 (2006), [URL] https://ideas.repec.org/p/crs/wpaper/2006-31.html</td> </tr><tr> <td>[64]</td> <td class="space">Gustafsson, J.; Hagmann, M.; Nielsen, J. P.; Scaillet, O., Local transformation kernel density estimation of loss distributions, J. Bus. Econom. Statist., 27, 2, 161-175 (2009), MR2516437</td> </tr><tr> <td>[65]</td> <td class="space">Hagmann, M.; Scaillet, O., Local multiplicative bias correction for asymmetric kernel density estimators, J. Econometrics, 141, 1, 213-249 (2007), MR2411743 · <a href="/1418.62136" class="nowrap">Zbl 1418.62136</a></td> </tr><tr> <td>[66]</td> <td class="space">Hall, P., Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. Multivariate Anal., 14, 1, 1-16 (1984), MR734096 · <a href="/0528.62028" class="nowrap">Zbl 0528.62028</a></td> </tr><tr> <td>[67]</td> <td class="space">Hall, P.; Wand, M. P., Minimizing $L_1$ distance in nonparametric density estimation, J. Multivariate Anal., 26, 1, 59-88 (1988), MR955204 · <a href="/0673.62030" class="nowrap">Zbl 0673.62030</a></td> </tr><tr> <td>[68]</td> <td class="space">Hanebeck, A., Nonparametric Distribution Function Estimation, 121 (2020), Karlsruher Institut für Technologie, [URL] https://core.ac.uk/download/pdf/326703853.pdf</td> </tr><tr> <td>[69]</td> <td class="space">Hanebeck, A.; Klar, B., Smooth distribution function estimation for lifetime distributions using Szasz-Mirakyan operators, Ann. Inst. Stat. Math., 19 (2021)</td> </tr><tr> <td>[70]</td> <td class="space">Hanif, M., Local linear estimation of jump-diffusion models by using asymmetric kernels, Stoch. Anal. Appl., 31, 6, 956-974 (2013), MR3175804 · <a href="/1283.62172" class="nowrap">Zbl 1283.62172</a></td> </tr><tr> <td>[71]</td> <td class="space">Harfouche, L.; Adjabi, S.; Zougab, N.; Funke, B., Multiplicative bias correction for discrete kernels, Stat. Methods Appl., 27, 2, 253-276 (2018), MR3807369 · <a href="/1396.62072" class="nowrap">Zbl 1396.62072</a></td> </tr><tr> <td>[72]</td> <td class="space">Harfouche, L.; Zougab, N.; Adjabi, S., Multivariate generalised gamma kernel density estimators and application to non-negative data, Int. J. Comput. Sci. Math., 11, 2, 137-157 (2020), MR4086907 · <a href="/1453.62443" class="nowrap">Zbl 1453.62443</a></td> </tr><tr> <td>[73]</td> <td class="space">Hirukawa, M., Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval, Comput. Statist. Data Anal., 54, 2, 473-495 (2010), MR2756441 · <a href="/1464.62090" class="nowrap">Zbl 1464.62090</a></td> </tr><tr> <td>[74]</td> <td class="space">Hirukawa, M., (Asymmetric Kernel Smoothing. Asymmetric Kernel Smoothing, SpringerBriefs in Statistics (2018), Springer, Singapore), xii+110, MR3821525 · <a href="/1401.62008" class="nowrap">Zbl 1401.62008</a></td> </tr><tr> <td>[75]</td> <td class="space">Hirukawa, M.; Murtazashvili, I.; Prokhorov, A., Uniform convergence rates for nonparametric estimators smoothed by the beta kernel, 36 (2020), Preprint, https://www.econ.ryukoku.ac.jp/ hirukawa/upload/uniform_31dec20_final.pdf</td> </tr><tr> <td>[76]</td> <td class="space">Hirukawa, M.; Sakudo, M., Nonnegative bias reduction methods for density estimation using asymmetric kernels, Comput. Statist. Data Anal., 75, 112-123 (2014), MR3178361 · <a href="/1506.62081" class="nowrap">Zbl 1506.62081</a></td> </tr><tr> <td>[77]</td> <td class="space">Hirukawa, M.; Sakudo, M., Family of the generalised gamma kernels: a generator of asymmetric kernels for nonnegative data, J. Nonparametr. Stat., 27, 1, 41-63 (2015), MR3304359 · <a href="/1320.62080" class="nowrap">Zbl 1320.62080</a></td> </tr><tr> <td>[78]</td> <td class="space">Hirukawa, M.; Sakudo, M., Testing symmetry of unknown densities via smoothing with the generalized gamma kernels, Econometrics, 4, 28, 27 (2016)</td> </tr><tr> <td>[79]</td> <td class="space">Hirukawa, M.; Sakudo, M., Another bias correction for asymmetric kernel density estimation with a parametric start, Statist. Probab. Lett., 145, 158-165 (2019), MR3873902 · <a href="/1407.62120" class="nowrap">Zbl 1407.62120</a></td> </tr><tr> <td>[80]</td> <td class="space">Hjort, N. L.; Glad, I. K., Nonparametric density estimation with a parametric start, Ann. Statist., 23, 3, 882-904 (1995), MR1345205 · <a href="/0838.62027" class="nowrap">Zbl 0838.62027</a></td> </tr><tr> <td>[81]</td> <td class="space">Hoang, D. H.; Pereira, L.; Kupka, N.; Tolosana-Delgado, R.; Frenzel, M.; Rudolph, M.; Gutzmer, J., Automated mineralogy particle dataset: apatite flotation (2020)</td> </tr><tr> <td>[82]</td> <td class="space">Hoffmann, T.; Jones, N., Unified treatment of the asymptotics of asymmetric kernel density estimators, 1-16 (2015), Preprint, arXiv:1512.03188</td> </tr><tr> <td>[83]</td> <td class="space">Hurvich, C. M., Data-driven choice of a spectrum estimate: extending the applicability of cross-validation methods, J. Amer. Statist. Assoc., 80, 392, 933-940 (1985), MR819597 · <a href="/0582.62082" class="nowrap">Zbl 0582.62082</a></td> </tr><tr> <td>[84]</td> <td class="space">Igarashi, G., Bias reductions for beta kernel estimation, J. Nonparametr. Stat., 28, 1, 1-30 (2016), MR3463548 · <a href="/1343.62016" class="nowrap">Zbl 1343.62016</a></td> </tr><tr> <td>[85]</td> <td class="space">Igarashi, G., Weighted log-normal kernel density estimation, Comm. Statist. Theory Methods, 45, 22, 6670-6687 (2016), MR3540109 · <a href="/1349.62116" class="nowrap">Zbl 1349.62116</a></td> </tr><tr> <td>[86]</td> <td class="space">Igarashi, G., Multivariate density estimation using a multivariate weighted log-normal kernel, Sankhya A, 80, 2, 247-266 (2018), MR3850066</td> </tr><tr> <td>[87]</td> <td class="space">Igarashi, G., Nonparametric direct density ratio estimation using beta kernel, Statistics, 54, 2, 257-280 (2020), MR4076244 · <a href="/1435.62123" class="nowrap">Zbl 1435.62123</a></td> </tr><tr> <td>[88]</td> <td class="space">Igarashi, G.; Kakizawa, Y., On improving convergence rate of Bernstein polynomial density estimator, J. Nonparametr. Stat., 26, 1, 61-84 (2014), MR3174309 · <a href="/1359.62117" class="nowrap">Zbl 1359.62117</a></td> </tr><tr> <td>[89]</td> <td class="space">Igarashi, G.; Kakizawa, Y., Re-formulation of inverse Gaussian, reciprocal inverse Gaussian, and Birnbaum-Saunders kernel estimators, Statist. Probab. Lett., 84, 235-246 (2014), MR3131281 · <a href="/1463.62115" class="nowrap">Zbl 1463.62115</a></td> </tr><tr> <td>[90]</td> <td class="space">Igarashi, G.; Kakizawa, Y., Bias corrections for some asymmetric kernel estimators, J. Statist. Plann. Inference, 159, 37-63 (2015), MR3299088 · <a href="/1311.62055" class="nowrap">Zbl 1311.62055</a></td> </tr><tr> <td>[91]</td> <td class="space">Igarashi, G.; Kakizawa, Y., Generalised gamma kernel density estimation for nonnegative data and its bias reduction, J. Nonparametr. Stat., 30, 3, 598-639 (2018), MR3843043 · <a href="/1404.62032" class="nowrap">Zbl 1404.62032</a></td> </tr><tr> <td>[92]</td> <td class="space">Igarashi, G.; Kakizawa, Y., Limiting bias-reduced Amoroso kernel density estimators for non-negative data, Comm. Statist. Theory Methods, 47, 20, 4905-4937 (2018), MR3833873 · <a href="/1508.62085" class="nowrap">Zbl 1508.62085</a></td> </tr><tr> <td>[93]</td> <td class="space">Igarashi, G.; Kakizawa, Y., Higher-order bias corrections for kernel type density estimators on the unit or semi-infinite interval, J. Nonparametr. Stat., 32, 3, 617-647 (2020), MR4136585 · <a href="/1465.62071" class="nowrap">Zbl 1465.62071</a></td> </tr><tr> <td>[94]</td> <td class="space">Igarashi, G.; Kakizawa, Y., Multiplicative bias correction for asymmetric kernel density estimators revisited, Comput. Statist. Data Anal., 141, 40-61 (2020), MR3979322 · <a href="/1507.62081" class="nowrap">Zbl 1507.62081</a></td> </tr><tr> <td>[95]</td> <td class="space">Jeon, Y.; Kim, J. H.T., A gamma kernel density estimation for insurance loss data, Insurance Math. Econom., 53, 3, 569-579 (2013), MR3130451 · <a href="/1290.62099" class="nowrap">Zbl 1290.62099</a></td> </tr><tr> <td>[96]</td> <td class="space">Jin, X.; Kawczak, J., Birnbaum-Saunders and lognormal kernel estimators for modelling durations in high frequency financial data, Ann. Econ. Finance, 4, 1, 103-124 (2003), [URL] http://aeconf.com/Articles/May2003/aef040106.pdf</td> </tr><tr> <td>[97]</td> <td class="space">Jones, M. C., Simple boundary correction for kernel density estimation, Stat. Comput., 3, 135-146 (1993)</td> </tr><tr> <td>[98]</td> <td class="space">Jones, M. C.; Foster, P. J., Generalized jackknifing and higher order kernels, J. Nonparametr. Stat., 3, 1, 81-94 (1993), MR1272163 · <a href="/1383.62098" class="nowrap">Zbl 1383.62098</a></td> </tr><tr> <td>[99]</td> <td class="space">Jones, M. C.; Foster, P. J., A simple nonnegative boundary correction method for kernel density estimation, Statist. Sinica, 6, 4, 1005-1013 (1996), MR1422417 · <a href="/0859.62037" class="nowrap">Zbl 0859.62037</a></td> </tr><tr> <td>[100]</td> <td class="space">Jones, M. C.; Henderson, D. A., Kernel-type density estimation on the unit interval, Biometrika, 94, 4, 977-984 (2007), MR2416803 · <a href="/1156.62026" class="nowrap">Zbl 1156.62026</a></td> </tr><tr> <td>[101]</td> <td class="space">Jones, M. C.; Linton, O.; Nielsen, J. P., A simple bias reduction method for density estimation, Biometrika, 82, 2, 327-338 (1995), MR1354232 · <a href="/0823.62033" class="nowrap">Zbl 0823.62033</a></td> </tr><tr> <td>[102]</td> <td class="space">Kakizawa, Y., Bernstein polynomial probability density estimation, J. Nonparametr. Stat., 16, 5, 709-729 (2004), MR2068610 · <a href="/1080.62019" class="nowrap">Zbl 1080.62019</a></td> </tr><tr> <td>[103]</td> <td class="space">Kakizawa, Y., Nonparametric density estimation for nonnegative data, using symmetrical-based inverse and reciprocal inverse Gaussian kernels through dual transformation, J. Statist. Plann. Inference, 193, 117-135 (2018), MR3713468 · <a href="/1377.62108" class="nowrap">Zbl 1377.62108</a></td> </tr><tr> <td>[104]</td> <td class="space">Kakizawa, Y., Multivariate non-central Birnbaum-Saunders kernel density estimator for nonnegative data, J. Statist. Plann. Inference, 209, 187-207 (2020), MR4096263 · <a href="/1441.62092" class="nowrap">Zbl 1441.62092</a></td> </tr><tr> <td>[105]</td> <td class="space">Kakizawa, Y., A class of Birnbaum-Saunders type kernel density estimators for nonnegative data, Comput. Statist. Data Anal., 161, Article 107249 pp. (2021), 18pp. MR4244643 · <a href="/1543.62306" class="nowrap">Zbl 1543.62306</a></td> </tr><tr> <td>[106]</td> <td class="space">Kakizawa, Y., Recursive asymmetric kernel density estimation for nonnegative data, J. Nonparametr. Stat., 33, 2, 197-224 (2021), MR4279948 · <a href="/1472.62055" class="nowrap">Zbl 1472.62055</a></td> </tr><tr> <td>[107]</td> <td class="space">Kakizawa, Y.; Igarashi, G., Inverse gamma kernel density estimation for nonnegative data, J. Korean Statist. Soc., 46, 2, 194-207 (2017), MR3648359 · <a href="/1362.62080" class="nowrap">Zbl 1362.62080</a></td> </tr><tr> <td>[108]</td> <td class="space">Kokonendji, C. C.; Libengué Dobélé-Kpoka, F. G.B., Asymptotic results for continuous associated kernel estimators of density functions, Afr. Diaspora J. Math., 21, 1, 87-97 (2018), MR3885552 · <a href="/1409.62084" class="nowrap">Zbl 1409.62084</a></td> </tr><tr> <td>[109]</td> <td class="space">Kokonendji, C. C.; Senga Kiessé, T., Discrete associated kernels method and extensions, Stat. Methodol., 8, 6, 497-516 (2011), MR2834036 · <a href="/1248.62052" class="nowrap">Zbl 1248.62052</a></td> </tr><tr> <td>[110]</td> <td class="space">Kokonendji, C. C.; Senga Kiessé, T.; Balakrishnan, N., Semiparametric estimation for count data through weighted distributions, J. Statist. Plann. Inference, 139, 10, 3625-3638 (2009), MR2549110 · <a href="/1168.62032" class="nowrap">Zbl 1168.62032</a></td> </tr><tr> <td>[111]</td> <td class="space">Kokonendji, C. C.; Somé, S. M., On multivariate associated kernels to estimate general density functions, J. Korean Statist. Soc., 47, 1, 112-126 (2018), MR3760293 · <a href="/1390.62055" class="nowrap">Zbl 1390.62055</a></td> </tr><tr> <td>[112]</td> <td class="space">Kokonendji, C. C.; Somé, S. M., Bayesian bandwidths in semiparametric modelling for nonnegative orthant data with diagnostics, Stats, 4, 162-183 (2021)</td> </tr><tr> <td>[113]</td> <td class="space">Kokonendji, C. C.; Varron, D., Performance of discrete associated kernel estimators through the total variation distance, Statist. Probab. Lett., 110, 225-235 (2016), MR3474762 · <a href="/1338.62107" class="nowrap">Zbl 1338.62107</a></td> </tr><tr> <td>[114]</td> <td class="space">Koul, H. L.; Song, W., Large sample results for varying kernel regression estimates, J. Nonparametr. Stat., 25, 4, 829-853 (2013), MR3174299 · <a href="/1416.62199" class="nowrap">Zbl 1416.62199</a></td> </tr><tr> <td>[115]</td> <td class="space">Kristensen, D., Nonparametric filtering of the realized spot volatility: a kernel-based approach, Econom. Theory, 26, 1, 60-93 (2010), MR2587103 · <a href="/1183.91189" class="nowrap">Zbl 1183.91189</a></td> </tr><tr> <td>[116]</td> <td class="space">Kulasekera, K. B.; Padgett, W. J., Bayes bandwidth selection in kernel density estimation with censored data, J. Nonparametr. Stat., 18, 2, 129-143 (2006), MR2229885 · <a href="/1099.62037" class="nowrap">Zbl 1099.62037</a></td> </tr><tr> <td>[117]</td> <td class="space">Kuruwita, C. N.; Kulasekera, K. B.; Padgett, W. J., Density estimation using asymmetric kernels and Bayes bandwidths with censored data, J. Statist. Plann. Inference, 140, 7, 1765-1774 (2010), MR2606717 · <a href="/1184.62059" class="nowrap">Zbl 1184.62059</a></td> </tr><tr> <td>[118]</td> <td class="space">Leblanc, A., A bias-reduced approach to density estimation using Bernstein polynomials, J. Nonparametr. Stat., 22, 3-4, 459-475 (2010), MR2662607 · <a href="/1189.62058" class="nowrap">Zbl 1189.62058</a></td> </tr><tr> <td>[119]</td> <td class="space">Leblanc, A., On the boundary properties of Bernstein polynomial estimators of density and distribution functions, J. Statist. Plann. Inference, 142, 10, 2762-2778 (2012), MR2925964 · <a href="/1428.62144" class="nowrap">Zbl 1428.62144</a></td> </tr><tr> <td>[120]</td> <td class="space">Lejeune, M.; Sarda, P., Smooth estimators of distribution and density functions, Comput. Statist. Data Anal., 14, 4, 457-471 (1992), MR1192215 · <a href="/0937.62581" class="nowrap">Zbl 0937.62581</a></td> </tr><tr> <td>[121]</td> <td class="space">Lepskiĭ, O. V., Asymptotically minimax adaptive estimation. I. Upper bounds. optimally adaptive estimates, Teor. Veroyatn. Primen., 36, 4, 645-659 (1991), MR1147167 · <a href="/0738.62045" class="nowrap">Zbl 0738.62045</a></td> </tr><tr> <td>[122]</td> <td class="space">Li, X.; Xiao, J.; Shi, J., Statistical inference in the partial linear models with the inverse Gaussian kernel, Comm. Statist. Simulation Comput., 48, 1, 240-263 (2019), MR3937087 · <a href="/07551431" class="nowrap">Zbl 07551431</a></td> </tr><tr> <td>[123]</td> <td class="space">Li, X.; Xiao, J.; Song, W.; Shi, J., Local linear regression with reciprocal inverse Gaussian kernel, Metrika, 82, 6, 733-758 (2019), MR3975162 · <a href="/1434.62153" class="nowrap">Zbl 1434.62153</a></td> </tr><tr> <td>[124]</td> <td class="space">Libengué Dobélé-Kpoka, F. G.B.; Kokonendji, C. C., The mode-dispersion approach for constructing continuous associated kernels, Afr. Stat., 12, 3, 1417-1446 (2017), MR3743306 · <a href="/1392.62103" class="nowrap">Zbl 1392.62103</a></td> </tr><tr> <td>[125]</td> <td class="space">Liu, B.; Ghosh, S. K., On empirical estimation of mode based on weakly dependent samples, Comput. Statist. Data Anal., 152, Article 107046 pp. (2020), 21pp. MR4130895 · <a href="/1510.62166" class="nowrap">Zbl 1510.62166</a></td> </tr><tr> <td>[126]</td> <td class="space">Lu, L., On the uniform consistency of the Bernstein density estimator, Statist. Probab. Lett., 107, 52-61 (2015), MR3412755 · <a href="/1328.62236" class="nowrap">Zbl 1328.62236</a></td> </tr><tr> <td>[127]</td> <td class="space">Ma, X., On Gamma Kernel Function in Recursive Density Estimation, 66 (2019), Mississippi State University, [URL] https://hdl.handle.net/11668/14483</td> </tr><tr> <td>[128]</td> <td class="space">Malec, P.; Schienle, M., Nonparametric kernel density estimation near the boundary, Comput. Statist. Data Anal., 72, 57-76 (2014), MR3139348 · <a href="/1506.62122" class="nowrap">Zbl 1506.62122</a></td> </tr><tr> <td>[129]</td> <td class="space">Manivong, P., Estimating Multinomial Cell Probabilities Using Normalized Beta Kernels, 111 (2009), University of Manitoba, [URL] http://hdl.handle.net/1993/21676</td> </tr><tr> <td>[130]</td> <td class="space">Marchant, C.; Bertin, K.; Leiva, V.; Saulo, H., Generalized Birnbaum-Saunders kernel density estimators and an analysis of financial data, Comput. Statist. Data Anal., 63, 1-15 (2013), MR3040246 · <a href="/1468.62133" class="nowrap">Zbl 1468.62133</a></td> </tr><tr> <td>[131]</td> <td class="space">Markovich, L. A., Gamma kernel estimation of the density derivative on the positive semi-axis by dependent data, REVSTAT, 14, 3, 327-348 (2016), MR3520774 · <a href="/1369.62067" class="nowrap">Zbl 1369.62067</a></td> </tr><tr> <td>[132]</td> <td class="space">Markovich, L. A., Gamma kernel estimates for multivariate density and its partial derivative with respect to dependent data [in Russian], Fundam. Prikl. Mat., 22, 3, 145-177 (2018), MR3962366</td> </tr><tr> <td>[133]</td> <td class="space">Markovich, L., Light- and heavy-tailed density estimation by gamma-Weibull kernel, (Nonparametric Statistics. Nonparametric Statistics, Springer Proc. Math. Stat., vol. 250 (2018), Springer, Cham), 145-158, MR3933005 · <a href="/1414.62162" class="nowrap">Zbl 1414.62162</a></td> </tr><tr> <td>[134]</td> <td class="space">Markovich, L. A., Nonparametric estimation of multivariate density and its derivative by dependent data using gamma kernels, J. Math. Sci., 254, 4, 550-573 (2021) · <a href="/1462.62232" class="nowrap">Zbl 1462.62232</a></td> </tr><tr> <td>[135]</td> <td class="space">Lafaye de Micheaux, P.; Ouimet, F., A study of seven asymmetric kernels for the estimation of cumulative distribution functions, 1-38 (2020), Preprint, arXiv:2011.14893</td> </tr><tr> <td>[136]</td> <td class="space">Minc, H.; Sathre, L., Some inequalities involving $( r ! )^{1 / r}$, Proc. Edinburgh Math. Soc. (2), 14, 41-46 (1964), MR162751 · <a href="/0124.01003" class="nowrap">Zbl 0124.01003</a></td> </tr><tr> <td>[137]</td> <td class="space">Mnatsakanov, R. M.; Ruymgaart, F. H., Moment density estimation for positive random variables, Statistics, 46, 2, 215-230 (2012), MR2903523 · <a href="/1241.62046" class="nowrap">Zbl 1241.62046</a></td> </tr><tr> <td>[138]</td> <td class="space">Mnatsakanov, R.; Sarkisian, K., Varying kernel density estimation on $\mathbb{R}_+$, Statist. Probab. Lett., 82, 7, 1337-1345 (2012), MR2929784 · <a href="/1489.62114" class="nowrap">Zbl 1489.62114</a></td> </tr><tr> <td>[139]</td> <td class="space">Mombeni, H. A.; Masouri, B.; Akhoond, M. R., Asymmetric kernels for boundary modification in distribution function estimation, REVSTAT, 1-27 (2019), Available online at https://www.ine.pt/revstat/pdf/Asymmetrickernelsforboundarymodificationindistributionfunctionestimation.pdf</td> </tr><tr> <td>[140]</td> <td class="space">Mousa, A. M.; Hassan, M. K.; Fathi, A., A new non parametric estimator for pdf based on inverse gamma distribution, Comm. Statist. Theory Methods, 45, 23, 7002-7010 (2016), MR3544186 · <a href="/1349.62119" class="nowrap">Zbl 1349.62119</a></td> </tr><tr> <td>[141]</td> <td class="space">Ng, K. W.; Tian, G.-L.; Tang, M.-L., (Dirichlet and Related Distributions. Dirichlet and Related Distributions, Wiley Series in Probability and Statistics (2011), John Wiley & Sons, Ltd.: John Wiley & Sons, Ltd. Chichester), xxvi+310, MR2830563 · <a href="/1234.60006" class="nowrap">Zbl 1234.60006</a></td> </tr><tr> <td>[142]</td> <td class="space">Ouimet, F., Asymptotic properties of Bernstein estimators on the simplex, J. Multivariate Anal., 185, Article 104784 pp. (2021), 20 pp. MR4287788 · <a href="/1470.62067" class="nowrap">Zbl 1470.62067</a></td> </tr><tr> <td>[143]</td> <td class="space">Ouimet, F., On the boundary properties of Bernstein estimators on the simplex, 1-11 (2021), Preprint, arXiv:2006.11756</td> </tr><tr> <td>[144]</td> <td class="space">Ouimet, F., On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators, J. Math. Anal. Appl., 499, 1, Article 125033 pp. (2021), 18pp. MR4213687 · <a href="/1462.62772" class="nowrap">Zbl 1462.62772</a></td> </tr><tr> <td>[145]</td> <td class="space">Ouimet, F., A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications, 1-12 (2021), Preprint, arXiv:2104.04882</td> </tr><tr> <td>[146]</td> <td class="space">Ouimet, F., A multivariate normal approximation for the Dirichlet density and some applications, Stat, 11, 1, e410, 12 pp. (2022) · <a href="/07853550" class="nowrap">Zbl 07853550</a></td> </tr><tr> <td>[147]</td> <td class="space">Pereira, L.; Frenzel, M.; Hoang, D. H.; Tolosana-Delgado, R.; Rudolph, M.; Gutzmer, J., Computing single-particle flotation kinetics using automated mineralogy data and machine learning, Miner. Eng., 170, 1-10 (2021), 107054, http://dx.doi.org/10.1016/j.mineng.2021.107054</td> </tr><tr> <td>[148]</td> <td class="space">Prakasa Rao, B. L.S., Nonparametric Functional Estimation, xiv+522 (1983), Academic Press: Academic Press New York, MR0740865 · <a href="/0542.62025" class="nowrap">Zbl 0542.62025</a></td> </tr><tr> <td>[149]</td> <td class="space">Renault, O.; Scaillet, O., On the way to recovery: A nonparametric bias free estimation of recovery rate densities, J. Bank. Financ., 28, 2915-2931 (2004)</td> </tr><tr> <td>[150]</td> <td class="space">Salha, R. B., Hazard rate function estimation using inverse Gaussian kernel, IUG J. Nat. Eng. Stud., 20, 1, 73-84 (2012)</td> </tr><tr> <td>[151]</td> <td class="space">Saulo, H.; Leiva, V.; Ziegelmann, F. A.; Marchant, C., A nonparametric method for estimating asymmetric densities based on skewed Birnbaum-Saunders distributions applied to environmental data, Stoch. Environ. Res. Risk Assess., 27, 1479-1491 (2013)</td> </tr><tr> <td>[152]</td> <td class="space">Scaillet, O., Density estimation using inverse and reciprocal inverse Gaussian kernels, J. Nonparametr. Stat., 16, 1-2, 217-226 (2004), MR2053071 · <a href="/1049.62038" class="nowrap">Zbl 1049.62038</a></td> </tr><tr> <td>[153]</td> <td class="space">Schach, E.; Buchmann, M.; Tolosana-Delgado, R.; Leißner, T.; Kern, M.; van den Boogaart, K. G.; Rudolph, M.; Peuker, U. A., Multidimensional characterization of separation processes – part 1: Introducing kernel methods and entropy in the context of mineral processing using SEM-based image analysis, Miner. Eng., 137, 78-86 (2019)</td> </tr><tr> <td>[154]</td> <td class="space">Schucany, W. R.; Sommers, J. P., Improvement of kernel type density estimators, J. Amer. Statist. Assoc., 72, 358, 420-423 (1977), MR448691 · <a href="/0369.62039" class="nowrap">Zbl 0369.62039</a></td> </tr><tr> <td>[155]</td> <td class="space">Schuster, E. F., Incorporating support constraints into nonparametric estimators of densities, Comm. Statist. Theory Methods, 14, 5, 1123-1136 (1985), MR797636 · <a href="/0585.62070" class="nowrap">Zbl 0585.62070</a></td> </tr><tr> <td>[156]</td> <td class="space">Scott, D. W., (Multivariate Density Estimation. Multivariate Density Estimation, Wiley Series in Probability and Statistics (2015), John Wiley & Sons, Inc., Hoboken, NJ), xviii+350, MR3329609 · <a href="/1311.62004" class="nowrap">Zbl 1311.62004</a></td> </tr><tr> <td>[157]</td> <td class="space">Scott, D. W.; Wand, M. P., Feasibility of multivariate density estimates, Biometrika, 78, 1, 197-205 (1991), MR1118245</td> </tr><tr> <td>[158]</td> <td class="space">Senga Kiessé, T.; Cuny, H. E., Discrete triangular associated kernel and bandwidth choices in semiparametric estimation for count data, J. Stat. Comput. Simul., 84, 8, 1813-1829 (2014), MR3215725 · <a href="/1453.62445" class="nowrap">Zbl 1453.62445</a></td> </tr><tr> <td>[159]</td> <td class="space">Serfling, R. J., (Approximation Theorems of Mathematical Statistics. Approximation Theorems of Mathematical Statistics, Wiley Series in Probability and Mathematical Statistics (1980), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York), xvi+371, MR0595165 · <a href="/1001.62005" class="nowrap">Zbl 1001.62005</a></td> </tr><tr> <td>[160]</td> <td class="space">Shi, J.; Song, W., Asymptotic results in gamma kernel regression, Comm. Statist. Theory Methods, 45, 12, 3489-3509 (2016), MR3494026 · <a href="/1342.62055" class="nowrap">Zbl 1342.62055</a></td> </tr><tr> <td>[161]</td> <td class="space">Somé, S. M.; Kokonendji, C. C., Effects of associated kernels in nonparametric multiple regressions, J. Stat. Theory Pract., 10, 2, 456-471 (2016), MR3499725 · <a href="/1420.62149" class="nowrap">Zbl 1420.62149</a></td> </tr><tr> <td>[162]</td> <td class="space">Somé, S. M.; Kokonendji, C. C., Bayesian selector of adaptive bandwidth for multivariate gamma kernel estimator on $[ 0 , \infty )^d$, J. Appl. Stat., 1-22 (2021)</td> </tr><tr> <td>[163]</td> <td class="space">Somé, S. M.; Kokonendji, C. C.; Ibrahim, M., Associated kernel discriminant analysis for multivariate mixed data, Electron. J. Appl. Stat. Anal., 9, 2, 385-399 (2016), MR3567789</td> </tr><tr> <td>[164]</td> <td class="space">Song, Y.; Hou, W.; Zhou, S., Variance reduction estimation for return models with jumps using gamma asymmetric kernels, Stud. Nonlinear Dyn. Econom., 23, 5, Article 20180001 pp. (2019) · <a href="/07675510" class="nowrap">Zbl 07675510</a></td> </tr><tr> <td>[165]</td> <td class="space">Stadtmüller, U., Asymptotic distributions of smoothed histograms, Metrika, 30, 3, 145-158 (1983), MR0726014 · <a href="/0519.62019" class="nowrap">Zbl 0519.62019</a></td> </tr><tr> <td>[166]</td> <td class="space">Stadtmüller, U., Asymptotic properties of nonparametric curve estimates, Period. Math. Hungar., 17, 2, 83-108 (1986), MR0858109 · <a href="/0578.62044" class="nowrap">Zbl 0578.62044</a></td> </tr><tr> <td>[167]</td> <td class="space">Star-Lack, J.; Sun, M.; Kaestner, A.; Hassanein, R.; Virshup, G.; Berkus, T.; Oelhafen, M., Efficient scatter correction using asymmetric kernels, Proc. SPIE, 7258, 12 (2009)</td> </tr><tr> <td>[168]</td> <td class="space">Steele, M., (Probability Theory and Combinatorial Optimization. Probability Theory and Combinatorial Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69 (1997), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA), viii+159, MR1422018 · <a href="/0916.90233" class="nowrap">Zbl 0916.90233</a></td> </tr><tr> <td>[169]</td> <td class="space">Tanabe, K.; Sagae, M., An exact Cholesky decomposition and the generalized inverse of the variance-covariance matrix of the multinomial distribution, with applications, J. R. Stat. Soc. Ser. B Stat. Methodol., 54, 1, 211-219 (1992), MR1157720 · <a href="/0777.62054" class="nowrap">Zbl 0777.62054</a></td> </tr><tr> <td>[170]</td> <td class="space">Tang, F.-X.; Yang, Y.-F., Research of color image segmentation algorithm based on asymmetric kernel density estimation, J. Comput. Methods Sci. Eng., 17, 455-462 (2017)</td> </tr><tr> <td>[171]</td> <td class="space">Tenbusch, A., Two-dimensional Bernstein polynomial density estimators, Metrika, 41, 3-4, 233-253 (1994), MR1293514 · <a href="/0804.62045" class="nowrap">Zbl 0804.62045</a></td> </tr><tr> <td>[172]</td> <td class="space">Terrell, G. R.; Scott, D. W., On improving convergence rates for nonnegative kernel density estimators, Ann. Statist., 8, 5, 1160-1163 (1980), MR585714 · <a href="/0459.62031" class="nowrap">Zbl 0459.62031</a></td> </tr><tr> <td>[173]</td> <td class="space">Tromp, K., Neue Wege für die Beurteilung der Aufbereitung von Steinkohlen, Glückauf, 6, 125-131 (1937)</td> </tr><tr> <td>[174]</td> <td class="space">Vitale, R. A., Bernstein polynomial approach to density function estimation, (Statistical Inference and Related Topics (1975), Academic Press, New York), 87-99, MR0397977 · <a href="/0326.62027" class="nowrap">Zbl 0326.62027</a></td> </tr><tr> <td>[175]</td> <td class="space">Wansouwé, W. E.; Kokonendji, C. C.; Kolyang, D. T., Nonparametric estimation for probability mass function with disake: an r package for discrete associated kernel estimators, ARIMA Rev. Afr. Rech. Inform. Math. Appl., 19, 1-23 (2015), MR3337328</td> </tr><tr> <td>[176]</td> <td class="space">Weglarczyk, S., Kernel density estimation and its application, (ITM Web of Conferences: XLVIII Seminar of Applied Mathematics, Vol. 23 (2018)), 8</td> </tr><tr> <td>[177]</td> <td class="space">Xiao, J.; Li, X.; Shi, J., Estimation in a semiparametric partially linear errors-in-variables model with inverse Gaussian kernel, Comm. Statist. Theory Methods, 48, 17, 4394-4424 (2019), MR3978111 · <a href="/1508.62089" class="nowrap">Zbl 1508.62089</a></td> </tr><tr> <td>[178]</td> <td class="space">Xu, S., Asymmetric kernel density estimation based on grouped data with applications to loss model, Comm. Statist. Simulation Comput., 43, 3, 657-672 (2014), MR3200997 · <a href="/1291.62088" class="nowrap">Zbl 1291.62088</a></td> </tr><tr> <td>[179]</td> <td class="space">Yilmaz, A., Kernel-based object tracking using asymmetric kernels with adaptive scale and orientation selection, Mach. Vis. Appl., 22, 255-268 (2011)</td> </tr><tr> <td>[180]</td> <td class="space">Yin, X.-F.; Hao, Z.-F., Adaptative kernel density estimation using beta kernel, (Proc. 6th Int. Conf. on Machine Learning and Cybernetics (2007)), 19-22</td> </tr><tr> <td>[181]</td> <td class="space">Yuan-ming, D.; Wei, W.; Yi-ning, L.; Guo-xuan, Z., Enhanced mean shift tracking algorithm based on evolutive asymmetric kernel, (International Conference on Multimedia Technology (2011)), 5394-5398</td> </tr><tr> <td>[182]</td> <td class="space">Zhang, S., A note on the performance of the gamma kernel estimators at the boundary, Statist. Probab. Lett., 80, 7-8, 548-557 (2010), MR2595129 · <a href="/1185.62075" class="nowrap">Zbl 1185.62075</a></td> </tr><tr> <td>[183]</td> <td class="space">Zhang, S.; Karunamuni, R. J., On kernel density estimation near endpoints, J. Statist. Plann. Inference, 70, 2, 301-316 (1998), MR1649872 · <a href="/0938.62037" class="nowrap">Zbl 0938.62037</a></td> </tr><tr> <td>[184]</td> <td class="space">Zhang, S.; Karunamuni, R. J., On nonparametric density estimation at the boundary, J. Nonparametr. Stat., 12, 2, 197-221 (2000), MR1752313 · <a href="/0944.62038" class="nowrap">Zbl 0944.62038</a></td> </tr><tr> <td>[185]</td> <td class="space">Zhang, S.; Karunamuni, R. J., Boundary performance of the beta kernel estimators, J. Nonparametr. Stat., 22, 1-2, 81-104 (2010), MR2598955 · <a href="/1184.62061" class="nowrap">Zbl 1184.62061</a></td> </tr><tr> <td>[186]</td> <td class="space">Ziane, Y.; Adjabi, S.; Zougab, N., Adaptive Bayesian bandwidth selection in asymmetric kernel density estimation for nonnegative heavy-tailed data, J. Appl. Stat., 42, 8, 1645-1658 (2015), MR3350456 · <a href="/1514.62986" class="nowrap">Zbl 1514.62986</a></td> </tr><tr> <td>[187]</td> <td class="space">Ziane, Y.; Zougab, N.; Adjabi, S., Birnbaum-Saunders power-exponential kernel density estimation and Bayes local bandwidth selection for nonnegative heavy tailed data, Comput. Statist., 33, 1, 299-318 (2018), MR3754719 · <a href="/1417.62069" class="nowrap">Zbl 1417.62069</a></td> </tr><tr> <td>[188]</td> <td class="space">Ziane, Y.; Zougab, N.; Adjabi, S., Body tail adaptive kernel density estimation for nonnegative heavy-tailed data, Monte Carlo Methods Appl., 27, 1, 57-69 (2021), MR4223858 · <a href="/1467.62059" class="nowrap">Zbl 1467.62059</a></td> </tr><tr> <td>[189]</td> <td class="space">Zougab, N.; Adjabi, S., Multiplicative bias correction for generalized Birnbaum-Saunders kernel density estimators and application to nonnegative heavy tailed data, J. Korean Statist. Soc., 45, 1, 51-63 (2016), MR3456321 · <a href="/1330.62184" class="nowrap">Zbl 1330.62184</a></td> </tr><tr> <td>[190]</td> <td class="space">Zougab, N.; Adjabi, S.; Kokonendji, C. C., Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach, J. Stat. Comput. Simul., 83, 12, 2219-2231 (2013), MR3169297 · <a href="/1453.62449" class="nowrap">Zbl 1453.62449</a></td> </tr><tr> <td>[191]</td> <td class="space">Zougab, N.; Adjabi, S.; Kokonendji, C. C., Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation, Comput. Statist. Data Anal., 75, 28-38 (2014), MR3178355 · <a href="/1506.62212" class="nowrap">Zbl 1506.62212</a></td> </tr><tr> <td>[192]</td> <td class="space">Zougab, N.; Adjabi, S.; Kokonendji, C. C., Comparison study to bandwidth selection in binomial kernel estimation using Bayesian approaches, J. Stat. Theory Pract., 10, 1, 133-153 (2016), MR3453033 · <a href="/1420.62166" class="nowrap">Zbl 1420.62166</a></td> </tr><tr> <td>[193]</td> <td class="space">Zougab, N.; Harfouche, L.; Ziane, Y.; Adjabi, S., Multivariate generalized Birnbaum-Saunders kernel density estimators, Comm. Statist. Theory Methods, 47, 18, 4534-4555 (2018), MR3819800 · <a href="/1508.62091" class="nowrap">Zbl 1508.62091</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 1480.62102 - zbMATH Open