On directional multiple-output quantile regression
AbstractThis paper sheds some new light on the multivariate (projectional) quantiles recently introduced in Kong and Mizera (2008). Contrary to the sophisticated set analysis used there, we adopt a more parametric approach and study the subgradient conditions associated with these quantiles. In this setup, we introduce Lagrange multipliers which can be interpreted in various interesting ways. We also link these quantiles with portfolio optimization and present an alternative proof that the resulting quantile regions coincide with the halfspace depth ones. Our proof makes the link between halfspace depth contours and univariate quantiles of projections more explicit and results into an exact computation of sample quantile regions (hence also of halfspace depth regions) from projectional quantiles. Throughout, we systematically consider the regression case, which was barely touched in Kong and Mizera (2008). Above all, we study the projectional regression quantile regions and compare them with those resulting from the approach considered in Hallin, Paindaveine, and Siman (2009).To gain in generality and to make the comparison between both concepts easier, we present a general framework for directional multivariate(regression) quantiles which includes both approaches as particular cases and is of interest in itself.
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Bibliographic InfoPaper provided by ULB -- Universite Libre de Bruxelles in its series Working Papers ECARES with number 2009_011.
Date of creation: 2009
Date of revision:
Publication status: Published by: ECARES
Multivariate quantile; Quantile regression; Multiple-output regression;
Other versions of this item:
- Paindaveine, Davy & Siman, Miroslav, 2011. "On directional multiple-output quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 193-212, February.
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- Paindaveine, Davy & Šiman, Miroslav, 2012. "Computing multiple-output regression quantile regions," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 840-853.
- Davy Paindaveine & Miroslav Šiman, 2012. "Computing multiple-output regression quantile regions from projection quantiles," Computational Statistics, Springer, vol. 27(1), pages 29-49, March.
- Marc Hallin & Zudi Lu & Davy Paindaveine & Miroslav Siman, 2012. "Local Constant and Local Bilinear Multiple-Output Quantile Regression," Working Papers ECARES ECARES 2012-003, ULB -- Universite Libre de Bruxelles.
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