Oja quantiles

• Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
  [nagy@karlin.mff.cuni.cz]

• ECARES and Department of Mathematics, Université libre de Bruxelles, Brussels, Belgium [Davy.Paindaveine@ulb.be]

Abstract

One of the most successful concepts of multivariate quantiles is the concept of spatial (or geometric) quantiles; see Chaudhuri [1996]. An important drawback of these quantiles, however, is that they are equivariant under rigid-body transformations but not under general affine transformations. This was addressed through an inner-standardization approach in Hettmansperger et al. [2002] in the special case of the median, then in Serfling [2010] for general quantiles. Still for general quantiles, affine-equivariant spatial quantiles were obtained through a transformation-retransformation approach in Chakraborty [2003]. However, such approaches are not tailored-made for spatial quantiles and could in principle be applied to any multivariate quantiles. Moreover, they also require quite arbitrary choices, for instance the choice of an affine-equivariant scatter matrix functional in Serfling [2010].

Now, in the special case of the median, it is well-known that affine equivariance can be achieved in an elegant way by minimizing the volumes of data-based simplices. The Oja [1983] median can be considered to provide an affine-equivariant version of the spatial median: not only do both medians share the same asymptotic behavior under spherically symmetric distributions but they also can be phrased in a common simplex-based construction; see Paindaveine [2022] and Dürre et al. [2022]. More precisely, the spatial and Oja medians are obtained for $\mathrm{\ell }=1$ and $\mathrm{\ell }=d$ in the class of $\mathrm{\ell }$-medians

$${\mu }_{\mathrm{\ell },P}:=\arg \underset{\mu \in {ℝ}^d}{\min }{\mathrm{E}}_P[W_{\mathrm{\ell }}(\mu )],$$

where $W_{\mathrm{\ell }}(x):=m_{\mathrm{\ell }}(\mathrm{Simpl}(X_1,\mathrm{\dots },X_{\mathrm{\ell }},x))$ involves a random sample from $P$; here, $m_{\mathrm{\ell }}(\mathrm{Simpl}(x_1,\mathrm{\dots },x_{\mathrm{\ell }},x_{\mathrm{\ell }+1}))$ stands for the usual $\mathrm{\ell }$-measure of the simplex in ${ℝ}^d$ with vertices $x_1,\mathrm{\dots },x_{\mathrm{\ell }},x_{\mathrm{\ell }+1}$. In this talk, we define Oja quantiles based on such $d$-dimensional simplices that provide affine-equivariant versions of spatial quantiles—in the same way the Oja median provides an affine-equivariant version of the spatial median. This also leads to an Oja depth that, unlike the simplicial volume depth from Zuo et al. [2000], achieves affine invariance without relying on an external standardization. We provide a systematic investigation of the properties of Oja quantiles, in line with what was recently done for spatial quantiles in Konen et al. [2022].