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Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth

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  • Marc Hallin
  • Davy Paindaveine
  • Miroslav Siman

Abstract

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.

Suggested Citation

  • 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.
  • Handle: RePEc:eca:wpaper:2008_042
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    References listed on IDEAS

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    1. Gilbert W. Bassett, 2004. "Pessimistic Portfolio Allocation and Choquet Expected Utility," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 2(4), pages 477-492.
    2. 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.
    3. Abdous, B. & Theodorescu, R., 1992. "Note on the spatial quantile of a random vector," Statistics & Probability Letters, Elsevier, vol. 13(4), pages 333-336, March.
    4. Cascos, Ignacio & López-Díaz, Miguel, 2005. "Integral trimmed regions," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 404-424, October.
    5. Willa W. Chen & Rohit S. Deo, 2004. "Power transformations to induce normality and their applications," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 117-130.
    6. 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.
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