Document Zbl 1486.94043

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Math.</a> <a href="/?q=in%3A322167" title="Articles in this Issue">2013, Article ID 848271, 10 p. (2013)</a>. </div> <div class="abstract">Summary: Recently, a series of divergence measures have emerged from information theory and statistics and numerous inequalities have been established among them. However, none of them are a metric in topology. In this paper, we propose a class of metric divergence measures, namely, $L_p(P \parallel Q), P \geq 1,$ and study their mathematical properties. We then study an important divergence measure widely used in credit scoring, called information value. In particular, we explore the mathematical reasoning of weight of evidence and suggest a better alternative to weight of evidence. Finally, we propose using $L_p(P \parallel Q)$ as alternatives to information value to overcome its disadvantages.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7132158">Cited in <strong>7</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A17" title="MSC2020">94A17</a> </td> <td class="space"> Measures of information, entropy </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A91G40" title="MSC2020">91G40</a> </td> <td class="space"> Credit risk </td> </tr></table> </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 1486.94043" data-ciurl="/ci/07132158" data-biburl="/bibtex/07132158.bib" data-amsurl="/amsrefs/07132158.bib" data-xmlurl="/xml/07132158.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07132158.pdf" title="Zbl 1486.94043 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/10.1155/2013/848271" aria-label="DOI for “Metric divergence measures and information value in credit scoring”" title="10.1155/2013/848271">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/3.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Taneja, I. J., Generalized relative information and information inequalities, Journal of Inequalities in Pure and Applied Mathematics, 5, 1, 1-19 (2004) · <a href="/1051.94009" class="nowrap">Zbl 1051.94009</a></td> </tr><tr> <td>[2]</td> <td class="space">Hellinger, E., Neue begründung der theorie der quadratischen formen von unendlichen vielen veränderlichen, Journal Für Die Reine und Angewandte Mathematik, 136, 210-271 (1909) · <a href="/40.0393.01" class="nowrap">JFM 40.0393.01</a></td> </tr><tr> <td>[3]</td> <td class="space">Shannon, C. E., A mathematical theory of communication, The Bell System Technical Journal, 27, 379-423 (1948) · <a href="/1154.94303" class="nowrap">Zbl 1154.94303</a></td> </tr><tr> <td>[4]</td> <td class="space">Kullback, S.; Leibler, R. A., On information and sufficiency, Annals of Mathematical Statistics, 22, 79-86 (1951) · <a href="/0042.38403" class="nowrap">Zbl 0042.38403</a> · <a href="https://doi.org/10.1214/aoms/1177729694" class="nowrap">doi:10.1214/aoms/1177729694</a></td> </tr><tr> <td>[5]</td> <td class="space">Jeffreys, H., An invariant form for the prior probability in estimation problems, Proceedings of the Royal Society, 186, 453-461 (1946) · <a href="/0063.03050" class="nowrap">Zbl 0063.03050</a> · <a href="https://doi.org/10.1098/rspa.1946.0056" class="nowrap">doi:10.1098/rspa.1946.0056</a></td> </tr><tr> <td>[6]</td> <td class="space">Topsøe, F., Some inequalities for information divergence and related measures of discrimination, Institute of Electrical and Electronics Engineers, 46, 4, 1602-1609 (2000) · <a href="/1003.94010" class="nowrap">Zbl 1003.94010</a> · <a href="https://doi.org/10.1109/18.850703" class="nowrap">doi:10.1109/18.850703</a></td> </tr><tr> <td>[7]</td> <td class="space">Dragomir, S. S.; Šunde, J.; Buşe, C., New inequalities for Jeffreys divergence measure, Tamsui Oxford Journal of Mathematical Sciences, 16, 2, 295-309 (2000) · <a href="/0994.94017" class="nowrap">Zbl 0994.94017</a></td> </tr><tr> <td>[8]</td> <td class="space">Pearson, K., On the criterion that a given system of deviations from the probable in the case of correlated system of variables is such that it can be reasonable supposed to have arisen from random sampling, Philosophical Magazine, 50, 157-172 (1992) · <a href="/31.0238.04" class="nowrap">JFM 31.0238.04</a></td> </tr><tr> <td>[9]</td> <td class="space">Sibson, R., Information radius, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 14, 2, 149-160 (1969) · <a href="/0186.53301" class="nowrap">Zbl 0186.53301</a></td> </tr><tr> <td>[10]</td> <td class="space">Burbea, J.; Rao, C. R., Entropy differential metric, distance and divergence measures in probability spaces: a unified approach, Journal of Multivariate Analysis, 12, 4, 575-596 (1982) · <a href="/0526.60015" class="nowrap">Zbl 0526.60015</a> · <a href="https://doi.org/10.1016/0047-259X(82)90065-3" class="nowrap">doi:10.1016/0047-259X(82)90065-3</a></td> </tr><tr> <td>[11]</td> <td class="space">Burbea, J.; Rao, C. R., On the convexity of some divergence measures based on entropy functions, Institute of Electrical and Electronics Engineers, 28, 3, 489-495 (1982) · <a href="/0479.94009" class="nowrap">Zbl 0479.94009</a> · <a href="https://doi.org/10.1109/TIT.1982.1056497" class="nowrap">doi:10.1109/TIT.1982.1056497</a></td> </tr><tr> <td>[12]</td> <td class="space">Taneja, I. J.; Hawkes, P. W., New developments in generalized information measures, Advances in Imaging and Electron Physics, 91, 37-136 (1995)</td> </tr><tr> <td>[13]</td> <td class="space">Cressie, N.; Read, T. R. C., Multinomial goodness-of-fit tests, Journal of the Royal Statistical Society B, 46, 3, 440-464 (1984) · <a href="/0571.62017" class="nowrap">Zbl 0571.62017</a></td> </tr><tr> <td>[14]</td> <td class="space">Taneja, I. J., Generalized Symmetric Divergence Measures and Inequalities, RGMIA Research Report Collection, 7, 4 (2004)</td> </tr><tr> <td>[15]</td> <td class="space">Cover, T. M.; Thomas, J. A., Elements of InformationTheory (1991), New York, NY, USA: John Wiley & Sons, New York, NY, USA · <a href="/0762.94001" class="nowrap">Zbl 0762.94001</a> · <a href="https://doi.org/10.1002/0471200611" class="nowrap">doi:10.1002/0471200611</a></td> </tr><tr> <td>[16]</td> <td class="space">da Costa, G. A. T. F.; TANEJA, I. J., Generalized Symmetric Divergence Measures and Metric Spaces (2011), Computing Research Repository</td> </tr><tr> <td>[17]</td> <td class="space">Siddiqi, N., Credit Risk Scorecards—Developing and Implementing Intelligent Credit Scoring (2006), John Wiley & Sons</td> </tr><tr> <td>[18]</td> <td class="space">Siddiqi, N., Credit Risk Scorecards—Development and Implementation Using SAS (2011), LULU</td> </tr><tr> <td>[19]</td> <td class="space">Downing, D.; Clark, J., Barron’s E-Z Statistics (2009), Barron’s Educational Series</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>

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