Multivariate quantiles and multiple-output regression quantiles: From L1 optimization to halfspace depth
AbstractA new multivariate concept of quantile, based on a directional version of Koenker and Bassettâs 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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Bibliographic InfoPaper provided by ULB -- Universite Libre de Bruxelles in its series ULB Institutional Repository with number 2013/127979.
Date of creation: Apr 2010
Date of revision:
Publication status: Published in: Annals of statistics (2010) v.38 nÂ° 2,p.635-669
Halfspace depth; Multivariate quantile; Quantile regression;
Other versions of this item:
- Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
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