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April 2010 Multivariate quantiles and multiple-output regression quantiles: From L1 optimization to halfspace depth
Marc Hallin, Davy Paindaveine, Miroslav Šiman
Ann. Statist. 38(2): 635-669 (April 2010). DOI: 10.1214/09-AOS723


A new multivariate concept of quantile, based on a directional version of Koenker and Bassett’s traditional regression quantiles, is introduced for multivariate location and multiple-output regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow for efficient depth contour computations by means of parametric linear programming, but also for transferring from the quantile to the depth universe such asymptotic results as Bahadur representations. Finally, linear programming duality opens the way to promising developments in depth-related multivariate rank-based inference.


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Marc Hallin. Davy Paindaveine. Miroslav Šiman. "Multivariate quantiles and multiple-output regression quantiles: From L1 optimization to halfspace depth." Ann. Statist. 38 (2) 635 - 669, April 2010.


Published: April 2010
First available in Project Euclid: 19 February 2010

zbMATH: 1183.62088
MathSciNet: MR2604670
Digital Object Identifier: 10.1214/09-AOS723

Primary: 62H05
Secondary: 62J05

Keywords: halfspace depth , Multivariate quantile , Quantile regression

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 2 • April 2010
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