{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:57:32Z","timestamp":1759838252041},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2012,9,29]],"date-time":"2012-09-29T00:00:00Z","timestamp":1348876800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2014,1]]},"DOI":"10.1007\/s11222-012-9352-6","type":"journal-article","created":{"date-parts":[[2012,9,30]],"date-time":"2012-09-30T12:24:34Z","timestamp":1349007874000},"page":"51-63","source":"Crossref","is-referenced-by-count":17,"title":["Computing projection depth and its associated estimators"],"prefix":"10.1007","volume":"24","author":[{"given":"Xiaohui","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yijun","family":"Zuo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2012,9,29]]},"reference":[{"key":"9352_CR1","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1145\/235815.235821","volume":"22","author":"C.B. Barber","year":"1996","unstructured":"Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 469\u2013483 (1996)","journal-title":"ACM Trans. Math. Softw."},{"key":"9352_CR2","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/PL00009389","volume":"20","author":"D. Bremner","year":"1998","unstructured":"Bremner, D., Fukuda, K., Marzetta, A.: Primal-dual methods for vertex and facet enumeration. Discrete Comput. Geom. 20, 333\u2013357 (1998)","journal-title":"Discrete Comput. Geom."},{"key":"9352_CR3","first-page":"1808","volume":"20","author":"D.L. Donoho","year":"1992","unstructured":"Donoho, D.L., Gasko, M.: Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Stat. 20, 1808\u20131827 (1992)","journal-title":"Ann. Stat."},{"key":"9352_CR4","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1145\/360680.360694","volume":"18","author":"R.W. Floyd","year":"1975","unstructured":"Floyd, R.W., Rivest, R.L.: Algorithm 489: Select. Commun. ACM 18, 173 (1975)","journal-title":"Commun. ACM"},{"key":"9352_CR5","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1214\/09-AOS723","volume":"38","author":"M. Halin","year":"2010","unstructured":"Halin, M., Paindaveine, D., \u0160iman, M.: Multivariate quantiles and multiple-output regression quantiles: From L1 optimization to halfspace depth. Ann. Stat. 38, 635\u2013669 (2010)","journal-title":"Ann. Stat."},{"key":"9352_CR6","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1080\/00401706.1984.10487956","volume":"26","author":"D.M. Hawkins","year":"1984","unstructured":"Hawkins, D.M., Bradu, D., Kass, G.V.: Location of several outliers in multiple regression data using elemental sets. Technometrics 26, 197\u2013208 (1984)","journal-title":"Technometrics"},{"key":"9352_CR7","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s11634-010-0066-3","volume":"4","author":"M. Hubert","year":"2010","unstructured":"Hubert, M., Van\u00a0der Veeken, S.: Robust classification for skewed data. Adv. Data Anal. Classif. 4, 239\u2013254 (2010)","journal-title":"Adv. Data Anal. Classif."},{"key":"9352_CR8","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1214\/aos\/1176347498","volume":"18","author":"R.Y. Liu","year":"1990","unstructured":"Liu, R.Y.: On a notion of data depth based on random simplices. Ann. Stat. 18, 191\u2013219 (1990)","journal-title":"Ann. Stat."},{"key":"9352_CR9","first-page":"279","volume-title":"L1-Statistical Analysis and Related Methods","author":"R.Y. Liu","year":"1992","unstructured":"Liu, R.Y.: Data depth and multivariate rank tests. In: Dodge, Y. (ed.) L1-Statistical Analysis and Related Methods, pp. 279\u2013294. North-Holland, Amsterdam (1992)"},{"key":"9352_CR10","unstructured":"Liu, X.H., Zuo, Y.J., Wang, Z.Z.: Exactly computing bivariate projection depth median and contours. Preprint (2011)"},{"key":"9352_CR11","doi-asserted-by":"crossref","first-page":"2500","DOI":"10.1016\/j.csda.2009.01.017","volume":"53","author":"K. Mosler","year":"2009","unstructured":"Mosler, K., Lange, T., Bazovkin, P.: Computing zonoid trimmed regions of dimension d>2. Comput. Stat. Data Anal. 53, 2500\u20132510 (2009)","journal-title":"Comput. Stat. Data Anal."},{"key":"9352_CR12","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.jmva.2010.08.004","volume":"102","author":"D. Paindaveine","year":"2011","unstructured":"Paindaveine, D., \u0160iman, M.: On directional multiple-output quantile regression. J. Multivar. Anal. 102, 193\u2013392 (2011)","journal-title":"J. Multivar. Anal."},{"key":"9352_CR13","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1016\/j.csda.2010.11.014","volume":"56","author":"D. Paindaveine","year":"2012","unstructured":"Paindaveine, D., \u0160iman, M.: Computing multiple-output regression quantile regions. Comput. Stat. Data Anal. 56, 840\u2013853 (2012a)","journal-title":"Comput. Stat. Data Anal."},{"key":"9352_CR14","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s00180-011-0231-y","volume":"27","author":"D. Paindaveine","year":"2012","unstructured":"Paindaveine, D., \u0160iman, M.: Computing multiple-output regression quantile regions from projection quantiles. Comput. Stat. 27, 29\u201349 (2012b)","journal-title":"Comput. Stat."},{"key":"9352_CR15","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1080\/01621459.1999.10474129","volume":"94","author":"P.J. Rousseeuw","year":"1999","unstructured":"Rousseeuw, P.J., Hubert, M.: Regression depth (with discussion). J.\u00a0Am. Stat. Assoc. 94, 388\u2013433 (1999)","journal-title":"J.\u00a0Am. Stat. Assoc."},{"key":"9352_CR16","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1002\/0471725382","volume-title":"Robust Regression and Outlier Detection","author":"P.J. Rousseeuw","year":"1987","unstructured":"Rousseeuw, P.J., Leroy, A.: Robust Regression and Outlier Detection, p. 99. Wiley, New York (1987)"},{"key":"9352_CR17","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/S0167-9473(96)00027-8","volume":"23","author":"I. Ruts","year":"1996","unstructured":"Ruts, I., Rousseeuw, P.J.: Computing depth contours of bivariate point clouds. Comput. Stat. Data Anal. 23, 153\u2013168 (1996)","journal-title":"Comput. Stat. Data Anal."},{"key":"9352_CR18","series-title":"DIMACS Series in Discrete Mathematics and Theoretical Computer Science","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/dimacs\/072\/01","volume-title":"Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications","author":"R. Serfling","year":"2006","unstructured":"Serfling, R.: Depth functions in nonparametric multivariate inference. In: Liu, R.Y., Serfling, R., Souvaine, D.L. (eds.) Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 72, pp. 1\u201316. American Mathematical Society, Providence (2006)"},{"key":"9352_CR19","unstructured":"Stahel, W.A.: Breakdown of covariance estimators. Research Report 31, Fachgruppe f\u00fcr Statistik, ETH, Z\u00fcrich (1981)"},{"key":"9352_CR20","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1287\/opre.13.6.1029","volume":"13","author":"K. Swarup","year":"1965","unstructured":"Swarup, K.: Linear fractional functionals programming. Oper. Res. 13, 1029\u20131036 (1965)","journal-title":"Oper. Res."},{"key":"9352_CR21","first-page":"523","volume-title":"Proceedings of the International Congress of Mathematicians","author":"J.W. Tukey","year":"1975","unstructured":"Tukey, J.W.: Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, pp. 523\u2013531. Cana. Math. Congress, Montreal (1975)"},{"key":"9352_CR22","doi-asserted-by":"crossref","first-page":"1460","DOI":"10.1214\/aos\/1065705115","volume":"31","author":"Y.J. Zuo","year":"2003","unstructured":"Zuo, Y.J.: Projection based depth functions and associated medians. Ann. Stat. 31, 1460\u20131490 (2003)","journal-title":"Ann. Stat."},{"key":"9352_CR23","doi-asserted-by":"crossref","first-page":"2211","DOI":"10.1214\/009053606000000713","volume":"34","author":"Y.J. Zuo","year":"2006","unstructured":"Zuo, Y.J.: Multidimensional trimming based on projection depth. Ann. Stat. 34, 2211\u20132251 (2006)","journal-title":"Ann. Stat."},{"key":"9352_CR24","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1214\/009053604000000922","volume":"33","author":"Y.J. Zuo","year":"2005","unstructured":"Zuo, Y.J., Cui, H.J.: Depth weighted scatter estimators. Ann. Stat. 33, 381\u2013413 (2005)","journal-title":"Ann. Stat."},{"key":"9352_CR25","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1016\/j.csda.2010.09.010","volume":"55","author":"Y.J. Zuo","year":"2011","unstructured":"Zuo, Y.J., Lai, S.Y.: Exact computation of bivariate projection depth and the Stahel-Donoho estimator. Comput. Stat. Data Anal. 55, 1173\u20131179 (2011)","journal-title":"Comput. Stat. Data Anal."},{"key":"9352_CR26","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1016218226","volume":"28","author":"Y.J. Zuo","year":"2000","unstructured":"Zuo, Y.J., Serfling, R.: General notions of statistical depth function. Ann. Stat. 28, 461\u2013482 (2000)","journal-title":"Ann. Stat."},{"key":"9352_CR27","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1214\/aos\/1079120133","volume":"32","author":"Y.J. Zuo","year":"2004","unstructured":"Zuo, Y.J., Cui, H.J., He, X.M.: On the Stahel-Donoho estimators and depth-weighted means for multivariate data. Ann. Stat. 32, 189\u2013218 (2004)","journal-title":"Ann. Stat."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-012-9352-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11222-012-9352-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-012-9352-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T10:17:26Z","timestamp":1559384246000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11222-012-9352-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,29]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["9352"],"URL":"https:\/\/doi.org\/10.1007\/s11222-012-9352-6","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,9,29]]}}}