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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p> <i><b>Robust Regression and Outlier Detection</b></i> is a book on <a href="/wiki/Robust_statistics" title="Robust statistics">robust statistics</a>, particularly focusing on the <a href="/wiki/Breakdown_point" class="mw-redirect" title="Breakdown point">breakdown point</a> of methods for <a href="/wiki/Robust_regression" title="Robust regression">robust regression</a>. It was written by <a href="/wiki/Peter_Rousseeuw" title="Peter Rousseeuw">Peter Rousseeuw</a> and Annick M. Leroy, and published in 1987 by Wiley. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background">Background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_Regression_and_Outlier_Detection&action=edit&section=1" title="Edit section: Background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:HRDiagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/HRDiagram.png/220px-HRDiagram.png" decoding="async" width="220" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/HRDiagram.png/330px-HRDiagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/HRDiagram.png/440px-HRDiagram.png 2x" data-file-width="720" data-file-height="820" /></a><figcaption>The <a href="/wiki/Hertzsprung%E2%80%93Russell_diagram" title="Hertzsprung–Russell diagram">Hertzsprung–Russell diagram</a> of stars plotted by luminosity and color. Robust regression methods can fit a curve to the <a href="/wiki/Main_sequence" title="Main sequence">main sequence</a>, the central curve in this diagram, without being strongly influenced by the groups of stars far from the main sequence.</figcaption></figure> <p><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a> is the problem of inferring a linear functional relationship between a <a href="/wiki/Dependent_and_independent_variables#Statistics" title="Dependent and independent variables">dependent variable</a> and one or more <a href="/wiki/Dependent_and_independent_variables#Statistics" title="Dependent and independent variables">independent variables</a>, from data sets where that relation has been obscured by noise. <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a> assumes that the data all lie near the fit line or plane, but depart from it by the addition of <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a> residual values. In contrast, robust regression methods work even when some of the data points are <a href="/wiki/Outlier" title="Outlier">outliers</a> that bear no relation to the fit line or plane, possibly because the data draws from a mixture of sources or possibly because an adversarial agent is trying to corrupt the data to cause the regression method to produce an inaccurate result.<sup id="cite_ref-piepel_1-0" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A typical application, discussed in the book, involves the <a href="/wiki/Hertzsprung%E2%80%93Russell_diagram" title="Hertzsprung–Russell diagram">Hertzsprung–Russell diagram</a> of star types, in which one wishes to fit a curve through the <a href="/wiki/Main_sequence" title="Main sequence">main sequence</a> of stars without the fit being thrown off by the outlying <a href="/wiki/Giant_star" title="Giant star">giant stars</a> and <a href="/wiki/White_dwarf" title="White dwarf">white dwarfs</a>.<sup id="cite_ref-sonnberger_2-0" class="reference"><a href="#cite_note-sonnberger-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <i>breakdown point</i> of a robust regression method is the fraction of outlying data that it can tolerate while remaining accurate. For this style of analysis, higher breakdown points are better.<sup id="cite_ref-piepel_1-1" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The breakdown point for ordinary least squares is near zero (a single outlier can make the fit become arbitrarily far from the remaining uncorrupted data)<sup id="cite_ref-sonnberger_2-1" class="reference"><a href="#cite_note-sonnberger-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> while some other methods have breakdown points as high as 50%.<sup id="cite_ref-piepel_1-2" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Although these methods require few assumptions about the data, and work well for data whose noise is not well understood, they may have somewhat lower efficiency than ordinary least squares (requiring more data for a given accuracy of fit) and their implementation may be complex and slow.<sup id="cite_ref-weisberg_3-0" class="reference"><a href="#cite_note-weisberg-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Topics">Topics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_Regression_and_Outlier_Detection&action=edit&section=2" title="Edit section: Topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The book has seven chapters.<sup id="cite_ref-piepel_1-3" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-laycock_4-0" class="reference"><a href="#cite_note-laycock-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The first is introductory; it describes <a href="/wiki/Simple_linear_regression" title="Simple linear regression">simple linear regression</a> (in which there is only one independent variable), discusses the possibility of outliers that corrupt either the dependent or the independent variable, provides examples in which outliers produce misleading results, defines the breakdown point, and briefly introduces several methods for robust simple regression, including <a href="/wiki/Repeated_median_regression" title="Repeated median regression">repeated median regression</a>.<sup id="cite_ref-piepel_1-4" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sonnberger_2-2" class="reference"><a href="#cite_note-sonnberger-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The second and third chapters analyze in more detail the least median of squares method for regression (in which one seeks a fit that minimizes the <a href="/wiki/Median" title="Median">median</a> of the squared <a href="/wiki/Errors_and_residuals" title="Errors and residuals">residuals</a>) and the <a href="/wiki/Least_trimmed_squares" title="Least trimmed squares">least trimmed squares</a> method (in which one seeks to minimize the sum of the squared residuals that are below the median). These two methods both have breakdown point 50% and can be applied for both simple regression (chapter two) and multivariate regression (chapter three).<sup id="cite_ref-piepel_1-5" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kafadar_5-0" class="reference"><a href="#cite_note-kafadar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Although the least median has an appealing geometric description (as finding a strip of minimum height containing half the data), its low efficiency leads to the recommendation that the least trimmed squares be used instead; least trimmed squares can also be interpreted as using the least median method to find and eliminate outliers and then using simple regression for the remaining data,<sup id="cite_ref-laycock_4-1" class="reference"><a href="#cite_note-laycock-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and approaches simple regression in its efficiency.<sup id="cite_ref-segre_6-0" class="reference"><a href="#cite_note-segre-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> As well as describing these methods and analyzing their statistical properties, these chapters also describe how to use the authors' software for implementing these methods.<sup id="cite_ref-piepel_1-6" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The third chapter also includes descriptions of some alternative estimators with high breakdown points.<sup id="cite_ref-yohai_7-0" class="reference"><a href="#cite_note-yohai-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The fourth chapter describes one-dimensional estimation of a <a href="/wiki/Location_parameter" title="Location parameter">location parameter</a> or <a href="/wiki/Central_tendency" title="Central tendency">central tendency</a> and its software implementation, and the fifth chapter goes into more detail about the <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> used by the software to compute these estimates efficiently. The sixth chapter concerns <a href="/wiki/Outlier_detection" class="mw-redirect" title="Outlier detection">outlier detection</a>, comparing methods for identifying data points as outliers based on robust statistics with other widely used methods, and the final chapter concerns higher-dimensional location problems as well as <a href="/wiki/Time_series" title="Time series">time series</a> analysis and problems of fitting an ellipsoid or <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> to data.<sup id="cite_ref-piepel_1-7" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-laycock_4-2" class="reference"><a href="#cite_note-laycock-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kafadar_5-1" class="reference"><a href="#cite_note-kafadar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-yohai_7-1" class="reference"><a href="#cite_note-yohai-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> As well as using the breakdown point to compare statistical methods, the book also looks at their <a href="/wiki/Equivariant_map" title="Equivariant map">equivariance</a>: for which families of data transformations does the fit for transformed data equal the transformed version of the fit for the original data?<sup id="cite_ref-segre_6-1" class="reference"><a href="#cite_note-segre-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In keeping with the book's focus on applications, it features many examples of analyses done using robust methods, comparing the resulting estimates with the estimates obtained by standard non-robust methods.<sup id="cite_ref-weisberg_3-1" class="reference"><a href="#cite_note-weisberg-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-yohai_7-2" class="reference"><a href="#cite_note-yohai-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Theoretical material is included, but set aside so that it can be easily skipped over by less theoretically inclined readers. The authors take the position that robust methods can be used both to check the applicability of ordinary regression (when the results of both methods agree) and to supplant them in cases where the results disagree.<sup id="cite_ref-kafadar_5-2" class="reference"><a href="#cite_note-kafadar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Audience_and_reception">Audience and reception</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_Regression_and_Outlier_Detection&action=edit&section=3" title="Edit section: Audience and reception"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The book is aimed at applied statisticians, with the goal of convincing them to use the robust methods that it describes.<sup id="cite_ref-piepel_1-8" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Unlike previous work in robust statistics, it makes robust methods both understandable by and (through its associated software) available to practitioners.<sup id="cite_ref-weisberg_3-2" class="reference"><a href="#cite_note-weisberg-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> No prior knowledge of robust statistics is required,<sup id="cite_ref-laycock_4-3" class="reference"><a href="#cite_note-laycock-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> although some background in basic statistical techniques is assumed.<sup id="cite_ref-kafadar_5-3" class="reference"><a href="#cite_note-kafadar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The book could also be used as a textbook,<sup id="cite_ref-kafadar_5-4" class="reference"><a href="#cite_note-kafadar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> although reviewer P. J. Laycock calls the possibility of such a use "bold and progressive"<sup id="cite_ref-laycock_4-4" class="reference"><a href="#cite_note-laycock-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and reviewers Seheult and Green point out that such a course would be unlikely to fit into British statistical curricula.<sup id="cite_ref-segre_6-2" class="reference"><a href="#cite_note-segre-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Reviewers Seheult and Green complain that too much of the book acts as a user guide to the authors' software, and should have been trimmed.<sup id="cite_ref-segre_6-3" class="reference"><a href="#cite_note-segre-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> However, reviewer Gregory F. Piepel writes that "the presentation is very good", and he recommends the book to any user of statistical methods.<sup id="cite_ref-piepel_1-9" class="reference"><a href="#cite_note-piepel-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> And, while suggesting the reordering of some material, <a href="/wiki/Karen_Kafadar" title="Karen Kafadar">Karen Kafadar</a> strongly recommends the book as a textbook for graduate students and a reference for professionals.<sup id="cite_ref-kafadar_5-5" class="reference"><a href="#cite_note-kafadar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> And reviewer A. C. Atkinson concisely summarizes the book as "interesting and important".<sup id="cite_ref-atkinson_8-0" class="reference"><a href="#cite_note-atkinson-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_books">Related books</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_Regression_and_Outlier_Detection&action=edit&section=4" title="Edit section: Related books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There have been multiple previous books on robust regression and outlier detection, including:<sup id="cite_ref-kafadar_5-6" class="reference"><a href="#cite_note-kafadar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-yohai_7-3" class="reference"><a href="#cite_note-yohai-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <ul><li><i>Identification of Outliers</i> by D. M. Hawkins (1980)</li> <li><i>Robust Statistics</i> by <a href="/wiki/Peter_J._Huber" title="Peter J. Huber">Peter J. Huber</a> (1981)</li> <li><i>Introduction to Robust and Quasi-Robust Statistical Methods</i> by W. J. J. Rey (1983)</li> <li><i>Understanding Robust and Exploratory Data Analysis</i> by David C. Hoaglin, <a href="/wiki/Frederick_Mosteller" title="Frederick Mosteller">Frederick Mosteller</a>, and <a href="/wiki/John_Tukey" title="John Tukey">John Tukey</a> (1983)</li> <li><i>Robust Statistics</i> by Hampel, Ronchetti, Rousseeuw, and Stahel (1986)</li></ul> <p>In comparison, <i>Robust Regression and Outlier Detection</i> combines both robustness and the detection of outliers.<sup id="cite_ref-kafadar_5-7" class="reference"><a href="#cite_note-kafadar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> It is less theoretical, more focused on data and software, and more focused on the breakdown point than on other measures of robustness.<sup id="cite_ref-yohai_7-4" class="reference"><a href="#cite_note-yohai-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Additionally, it is the first to highlight the importance of "leverage", the phenomenon that samples with outlying values of the independent variable can have a stronger influence on the fit than samples where the independent variable has a central value.<sup id="cite_ref-atkinson_8-1" class="reference"><a href="#cite_note-atkinson-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Robust_Regression_and_Outlier_Detection&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-piepel-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-piepel_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-piepel_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-piepel_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-piepel_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-piepel_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-piepel_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-piepel_1-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-piepel_1-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-piepel_1-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-piepel_1-9"><sup><i><b>j</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPiepel1989" class="citation cs2">Piepel, Gregory F. (May 1989), "Review of <i>Robust Regression and Outlier Detection</i>", <i>Technometrics</i>, <b>31</b> (2): <span class="nowrap">260–</span>261, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1268828">10.2307/1268828</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1268828">1268828</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Technometrics&rft.atitle=Review+of+Robust+Regression+and+Outlier+Detection&rft.volume=31&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E260-%3C%2Fspan%3E261&rft.date=1989-05&rft_id=info%3Adoi%2F10.2307%2F1268828&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1268828%23id-name%3DJSTOR&rft.aulast=Piepel&rft.aufirst=Gregory+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+Regression+and+Outlier+Detection" class="Z3988"></span></span> </li> <li id="cite_note-sonnberger-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-sonnberger_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sonnberger_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-sonnberger_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSonnberger1989" class="citation cs2">Sonnberger, Harold (July–September 1989), "Review of <i>Robust Regression and Outlier Detection</i>", <i>Journal of Applied Econometrics</i>, <b>4</b> (3): <span class="nowrap">309–</span>311, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fjae.3950040309">10.1002/jae.3950040309</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2096530">2096530</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Econometrics&rft.atitle=Review+of+Robust+Regression+and+Outlier+Detection&rft.volume=4&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E309-%3C%2Fspan%3E311&rft.date=1989-07%2F1989-09&rft_id=info%3Adoi%2F10.1002%2Fjae.3950040309&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2096530%23id-name%3DJSTOR&rft.aulast=Sonnberger&rft.aufirst=Harold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+Regression+and+Outlier+Detection" class="Z3988"></span></span> </li> <li id="cite_note-weisberg-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-weisberg_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-weisberg_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-weisberg_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisberg1989" class="citation cs2">Weisberg, Stanford (July–August 1989), "Review of <i>Robust Regression and Outlier Detection</i>", <i>American Scientist</i>, <b>77</b> (4): <span class="nowrap">402–</span>403, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27855903">27855903</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Scientist&rft.atitle=Review+of+Robust+Regression+and+Outlier+Detection&rft.volume=77&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E402-%3C%2Fspan%3E403&rft.date=1989-07%2F1989-08&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27855903%23id-name%3DJSTOR&rft.aulast=Weisberg&rft.aufirst=Stanford&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+Regression+and+Outlier+Detection" class="Z3988"></span></span> </li> <li id="cite_note-laycock-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-laycock_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-laycock_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-laycock_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-laycock_4-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-laycock_4-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaycock1989" class="citation cs2">Laycock, P. J. (1989), "Review of <i>Robust Regression and Outlier Detection</i>", <i>Journal of the Royal Statistical Society, Series D (The Statistician)</i>, <b>38</b> (2): 138, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2348319">10.2307/2348319</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2348319">2348319</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+D+%28The+Statistician%29&rft.atitle=Review+of+Robust+Regression+and+Outlier+Detection&rft.volume=38&rft.issue=2&rft.pages=138&rft.date=1989&rft_id=info%3Adoi%2F10.2307%2F2348319&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2348319%23id-name%3DJSTOR&rft.aulast=Laycock&rft.aufirst=P.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+Regression+and+Outlier+Detection" class="Z3988"></span></span> </li> <li id="cite_note-kafadar-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-kafadar_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-kafadar_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-kafadar_5-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-kafadar_5-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-kafadar_5-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-kafadar_5-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-kafadar_5-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-kafadar_5-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKafadar1989" class="citation cs2"><a href="/wiki/Karen_Kafadar" title="Karen Kafadar">Kafadar, Karen</a> (June 1989), "Review of <i>Robust Regression and Outlier Detection</i>", <i>Journal of the American Statistical Association</i>, <b>84</b> (406): <span class="nowrap">617–</span>618, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2289958">10.2307/2289958</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2289958">2289958</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Statistical+Association&rft.atitle=Review+of+Robust+Regression+and+Outlier+Detection&rft.volume=84&rft.issue=406&rft.pages=%3Cspan+class%3D%22nowrap%22%3E617-%3C%2Fspan%3E618&rft.date=1989-06&rft_id=info%3Adoi%2F10.2307%2F2289958&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2289958%23id-name%3DJSTOR&rft.aulast=Kafadar&rft.aufirst=Karen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+Regression+and+Outlier+Detection" class="Z3988"></span></span> </li> <li id="cite_note-segre-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-segre_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-segre_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-segre_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-segre_6-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeheultGreen1989" class="citation cs2">Seheult, A. H.; Green, P. J. (1989), "Review of <i>Robust Regression and Outlier Detection</i>", <i>Journal of the Royal Statistical Society, Series A (Statistics in Society)</i>, <b>152</b> (1): <span class="nowrap">133–</span>134, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2982847">10.2307/2982847</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2982847">2982847</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+A+%28Statistics+in+Society%29&rft.atitle=Review+of+Robust+Regression+and+Outlier+Detection&rft.volume=152&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E133-%3C%2Fspan%3E134&rft.date=1989&rft_id=info%3Adoi%2F10.2307%2F2982847&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2982847%23id-name%3DJSTOR&rft.aulast=Seheult&rft.aufirst=A.+H.&rft.au=Green%2C+P.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARobust+Regression+and+Outlier+Detection" class="Z3988"></span></span> </li> <li id="cite_note-yohai-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-yohai_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-yohai_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-yohai_7-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-yohai_7-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-yohai_7-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYohai1989" class="citation cs2">Yohai, V. J. 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