Halfspace Depth and Regression Depth Characterize the Empirical Distribution
For multivariate data, the halfspace depth function can be seen as a natural and affine equivariant generalization of the univariate empirical cdf. For any multivariate data set, we show that the resulting halfspace depth function completely determines the empirical distribution. We do this by actua...
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| Veröffentlicht in: | Journal of multivariate analysis 1999-04, Vol.69 (1), p.135-153 |
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| container_title | Journal of multivariate analysis |
| container_volume | 69 |
| creator | Struyf, Anja J Rousseeuw, Peter J |
| description | For multivariate data, the halfspace depth function can be seen as a natural and affine equivariant generalization of the univariate empirical cdf. For any multivariate data set, we show that the resulting halfspace depth function completely determines the empirical distribution. We do this by actually reconstructing the data points from the depth contours. The data need not be in general position. Moreover, we prove the same property for regression depth. |
| doi_str_mv | 10.1006/jmva.1998.1804 |
| format | Article |
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Moreover, we prove the same property for regression depth.</description><subject>Exact sciences and technology</subject><subject>Linear inference, regression</subject><subject>location depth</subject><subject>location depth, multivariate ranking, reconstruction algorithm, regression depth</subject><subject>Mathematics</subject><subject>multivariate ranking</subject><subject>Nonparametric inference</subject><subject>Probability and statistics</subject><subject>reconstruction algorithm</subject><subject>regression depth</subject><subject>Sciences and techniques of general use</subject><subject>Statistics</subject><issn>0047-259X</issn><issn>1095-7243</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1999</creationdate><recordtype>article</recordtype><sourceid>X2L</sourceid><recordid>eNp1kM1LxDAQxYMouK5ePffgtXWSNP04yu7qKguCKHgLaTK1WbbdktQF_etN6aInD5N5hPeGx4-QawoJBchut-1BJbQsi4QWkJ6QGYVSxDlL-SmZAaR5zET5fk4uvN8CUCrydEY2a7Wrfa80RkvshyZSnYle8MOh93bfHT8XjXJKD-jsN0ZDg9Gq7a2zWu2ipfWDs9XnENyX5KxWO49Xxz0nb_er18U63jw_PC7uNrHmPE9jmoNJOdNgIBUVZ0EBp0xpnWIlDC1QCzAlspqFQKYKbbSBinLIKlYUgs9JMt3Vbu-9w1r2zrbKfUkKcmQhRxZyZCFHFiHwNAUc9qh_3Yg4GjslD5KrrAzPV5iQG6UdZZh-3FxIKrhshjYcu5mO9coHArVTnbb-r0Ke89A02IrJhoHEwaKTXlvsNBrrUA_S7O1_dX8AUZOLvg</recordid><startdate>199904</startdate><enddate>199904</enddate><creator>Struyf, Anja J</creator><creator>Rousseeuw, Peter J</creator><general>Elsevier Inc</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>IQODW</scope><scope>DKI</scope><scope>X2L</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>199904</creationdate><title>Halfspace Depth and Regression Depth Characterize the Empirical Distribution</title><author>Struyf, Anja J ; Rousseeuw, Peter J</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c3374-170d432c0d045b322c00312acc4eb5d18ec50d9e2f2c336a8cdcd0b1306b28853</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1999</creationdate><topic>Exact sciences and technology</topic><topic>Linear inference, regression</topic><topic>location depth</topic><topic>location depth, multivariate ranking, reconstruction algorithm, regression depth</topic><topic>Mathematics</topic><topic>multivariate ranking</topic><topic>Nonparametric inference</topic><topic>Probability and statistics</topic><topic>reconstruction algorithm</topic><topic>regression depth</topic><topic>Sciences and techniques of general use</topic><topic>Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Struyf, Anja J</creatorcontrib><creatorcontrib>Rousseeuw, Peter J</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>RePEc IDEAS</collection><collection>RePEc</collection><collection>CrossRef</collection><jtitle>Journal of multivariate analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Struyf, Anja J</au><au>Rousseeuw, Peter J</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Halfspace Depth and Regression Depth Characterize the Empirical Distribution</atitle><jtitle>Journal of multivariate analysis</jtitle><date>1999-04</date><risdate>1999</risdate><volume>69</volume><issue>1</issue><spage>135</spage><epage>153</epage><pages>135-153</pages><issn>0047-259X</issn><eissn>1095-7243</eissn><coden>JMVAAI</coden><abstract>For multivariate data, the halfspace depth function can be seen as a natural and affine equivariant generalization of the univariate empirical cdf. 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| ispartof | Journal of multivariate analysis, 1999-04, Vol.69 (1), p.135-153 |
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| language | eng |
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| source | RePEc; Elsevier ScienceDirect Journals Complete; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Exact sciences and technology Linear inference, regression location depth location depth, multivariate ranking, reconstruction algorithm, regression depth Mathematics multivariate ranking Nonparametric inference Probability and statistics reconstruction algorithm regression depth Sciences and techniques of general use Statistics |
| title | Halfspace Depth and Regression Depth Characterize the Empirical Distribution |
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