Document Zbl 0955.62608

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With comments by Ronald A. Thisted and M. R. Osborne and a rejoinder by the authors.</strong> <i>(English)</i> <a class="label nowrap" href="/0955.62608">Zbl 0955.62608</a> </h2> <div class="source"> <a href="/serials/1097" title="Journal Profile">Stat. Sci.</a> <a href="/?q=in%3A73563" title="Articles in this Issue">12, No. 4, 279-300 (1997)</a>. </div> <div class="abstract">Summary: Since the time of Gauss, it has been generally accepted that $l_2$-methods of combining observations by minimizing sums of squared errors have significant computational advantages over earlier $l_1$-methods based on minimization of absolute errors advocated by Boscovich, Laplace and others. However, $l_1$-methods are known to have significant robustness advantages over $l_2$-methods in many applications, and related quantile regression methods provide a useful, complementary approach to classical least-squares estimation of statistical models. Combining recent advances in interior point methods for solving linear programs with a new statistical preprocessing approach for $l_1$-type problems, we obtain a 10- to 100-fold improvement in computational speeds over current (simplex-based) $l_1$-algorithms in large problems, demonstrating that $l_1$-methods can be made competitive with $l_2$-methods in terms of computational speed throughout the entire range of problem sizes. Formal complexity results suggest that $l_1$-regression can be made faster than least-squares regression for $n$ sufficiently large and $p$ modest.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A1561989">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A1561989">Cited in <strong>127</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62J05" title="MSC2020">62J05</a> </td> <td class="space"> Linear regression; mixed models </td> </tr></table> </div> <div class="software"> <h3>Software:</h3><a href="/software/29823">Algorithm 478</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 0955.62608" data-ciurl="/ci/01561989" data-biburl="/bibtex/01561989.bib" data-amsurl="/amsrefs/01561989.bib" data-xmlurl="/xml/01561989.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01561989.pdf" title="Zbl 0955.62608 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/10.1214/ss/1030037960" aria-label="DOI for “The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators. With comments by Ronald A. Thisted and M. R. Osborne and a rejoinder by the authors”" title="10.1214/ss/1030037960">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">ney, A., Ostrouchov, S. and Sorenson, D. (1995). LAPACK Users’ Guide. SIAM, Philadelphia.</td> </tr><tr> <td>[2]</td> <td class="space">Barrodale, I. and Roberts, F. D. K. (1974). Solution of an overdetermined sy stem of equations in the 1 norm. Communications of the ACM 17 319-320.</td> </tr><tr> <td>[3]</td> <td class="space">Bartels, R. and Conn, A. (1980). Linearly constrained discrete 1 problems. ACM Trans. Math. Software 6 594-608. · <a href="/0448.49017" class="nowrap">Zbl 0448.49017</a> · <a href="https://doi.org/10.1145/355921.355930" class="nowrap">doi:10.1145/355921.355930</a></td> </tr><tr> <td>[4]</td> <td class="space">Bloomfield, P. and Steiger, W. L. (1983). 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