Document Zbl 1203.62128

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We propose an $\ell _{1}$-penalized maximum likelihood estimator in an appropriate parameterization. This kind of estimation belongs to a class of problems where optimization and theory for non-convex functions is needed. This distinguishes itself very clearly from high-dimensional estimation with convex loss- or objective functions as, for example, with the Lasso in linear or generalized linear models. Mixture models represent a prime and important example where non-convexity arises. For FMR models, we develop an efficient EM algorithm for numerical optimization with provable convergence properties. Our penalized estimator is numerically better posed (e.g., boundedness of the criterion function) than unpenalized maximum likelihood estimation, and it allows for effective statistical regularization including variable selection. We also present some asymptotic theory and oracle inequalities: due to non-convexity of the negative log-likelihood function, different mathematical arguments are needed than for problems with convex losses. Finally, we apply the new method to both simulated and real data.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5840530">Cited in <strong>7</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5840530">Cited in <strong>101</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J12" title="MSC2020">62J12</a> </td> <td class="space"> Generalized linear models (logistic models) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62F12" title="MSC2020">62F12</a> </td> <td class="space"> Asymptotic properties of parametric estimators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J07" title="MSC2020">62J07</a> </td> <td class="space"> Ridge regression; shrinkage estimators (Lasso) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C90" title="MSC2020">90C90</a> </td> <td class="space"> Applications of mathematical programming </td> </tr><tr> <td class="mono"> 65C60 </td> <td class="space"> Computational problems in statistics (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aadaptive+lasso">adaptive lasso</a>; <a href="/?q=ut%3Afinite+mixture+models">finite mixture models</a>; <a href="/?q=ut%3Ageneralized+EM+algorithm">generalized EM algorithm</a>; <a href="/?q=ut%3Ahigh-dimensional+estimation">high-dimensional estimation</a>; <a href="/?q=ut%3Alasso">lasso</a>; <a href="/?q=ut%3Aoracle+inequality">oracle inequality</a></div> <div class="software"> <h3>Software:</h3><a href="/software/8159">glmnet</a>; <a href="/software/6087">flexmix</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 1203.62128" data-ciurl="/ci/05840530" data-biburl="/bibtex/05840530.bib" data-amsurl="/amsrefs/05840530.bib" data-xmlurl="/xml/05840530.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05840530.pdf" title="Zbl 1203.62128 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/s11749-010-0197-z" aria-label="DOI for “$\ell_{1}$-penalization for mixture regression models”" title="10.1007/s11749-010-0197-z">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1202.6046"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">Bertsekas D (1995) Nonlinear programming. 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