On a Geometric Notion of Quantiles for Multivariate Data

Abstract

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Multivariate Symmetry and Asymmetry

“…Such indices may be used to order distributions, for example "F is less asymmetric than G" if A F ≤ A G . Alternative asymmetry functionals are given in [33] and [18].…”

A Depth Function and a Scale Curve Based on Spatial Quantiles

“…In particular, the wellknown spatial median of F is Q F (0), which we shall also denote by M F . Unlike many notions of multivariate quantiles, it is relatively straightforward to extend spatial quantiles to the setting of Banach spaces, as discussed in Kemperman (1987) for the spatial median and by Chaudhuri (1996) for the general case.…”

“…The quantile Q F (u) always exists for any u, and it is unique if d ≥ 2 and F is not supported on a straight line (see Chaudhuri, 1996). Moreover, the spatial quantile function characterizes the associated distribution, in the sense that Q F = Q G implies F = G (see Koltchinskii, 1997, Cor.…”

“…These have no definitive multivariate generalization, however, and a variety of ad hoc notions of multivariate quantiles have been formulated (see Serfling, 2002b, for a partial review). Here we focus on a particular form of multivariate quantiles, the spatial quantiles, introduced by Chaudhuri (1996) and Koltchinskii (1997) as a certain form of generalization of the univariate case based on the L 1 norm, and on a particular descriptive measure, spread. In particular, we treat the spatial "volume functional", whose values are noted by Chaudhuri (1996) to provide multivariate analogues of such univariate measures as interquantile ranges.…”

Robust fusion methods for Big Data

“…where P is a probability in some Banach space (E, · ) and x ∈ E, introduced by Chaudhuri (1996), formulated (in a different way) by Vardi and Zhand (2000), and extended to a very general setup by Chakraborty and Chaudhuri (2014). We want to address the consistency, efficiency, robustness and computational time properties of the RFM proposals.…”