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 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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Bibliographic InfoPaper provided by ULB -- Universite Libre de Bruxelles in its series Working Papers ECARES with number 2008_042.
Date of creation: 2008
Date of revision:
Publication status: Published by: ECARES
Other versions of this item:
- Marc Hallin & Davy Paindaveine & Miroslav Šiman, 2010. "Multivariate quantiles and multiple-output regression quantiles: From L1 optimization to halfspace depth," ULB Institutional Repository 2013/127979, ULB -- Universite Libre de Bruxelles.
- NEP-ALL-2008-12-21 (All new papers)
- NEP-ECM-2008-12-21 (Econometrics)
- NEP-FDG-2008-12-21 (Financial Development & Growth)
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