On directional multiple-output quantile regression
This 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.
|Date of creation:||2009|
|Publication status:||Published by: ECARES|
|Contact details of provider:|| Postal: Av. F.D., Roosevelt, 39, 1050 Bruxelles|
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- Bertsimas, Dimitris & Lauprete, Geoffrey J. & Samarov, Alexander, 2004. "Shortfall as a risk measure: properties, optimization and applications," Journal of Economic Dynamics and Control, Elsevier, vol. 28(7), pages 1353-1381, April.
- 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.
- 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.
- Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, Diciembre.
- Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521608275.
- Roger Koenker & Kevin F. Hallock, 2001. "Quantile Regression," Journal of Economic Perspectives, American Economic Association, vol. 15(4), pages 143-156, Fall.
- Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
- Wei, Ying, 2008. "An Approach to Multivariate Covariate-Dependent Quantile Contours With Application to Bivariate Conditional Growth Charts," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 397-409, March.
- Robert Serfling, 2002. "Quantile functions for multivariate analysis: approaches and applications," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(2), pages 214-232.
- Jules Sadefo Kamdem, 2005. "Value-At-Risk And Expected Shortfall For Linear Portfolios With Elliptically Distributed Risk Factors," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(05), pages 537-551. Full references (including those not matched with items on IDEAS)