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Hallin, Marc; Paindaveine, Davy; Šiman, Miroslav

Multivariate quantiles and multiple-output regression quantiles: from \(L_{1}\) optimization to halfspace depth. (English) Zbl 1183.62088

Ann. Stat. 38, No. 2, 635-669 (2010).
Summary: A new multivariate concept of quantiles, based on a directional version of R. Koenker and G. Bassett’s [Econometrica 46, 33–50 (1978; Zbl 0373.62038)] 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 J. W. Tukey [Proc. Int. Congr. Math., Vancouver 1974, Vol. 2, 523–531 (1975; Zbl 0347.62002)]. 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.

Cited in 7 Reviews
Cited in 87 Documents

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G20 Asymptotic properties of nonparametric inference
90C05 Linear programming
62A09 Graphical methods in statistics
62J05 Linear regression; mixed models
90C90 Applications of mathematical programming

Keywords:

multivariate quantile; quantile regression; halfspace depth

Citations:

Zbl 0373.62038; Zbl 0347.62002

Software:

AS 307; quantreg
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Full Text: DOI arXiv

References:

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