Quantile tomography: using quantiles with multivariate data

Directional quantile envelopes—essentially, depth contours—are a possible way to condense thedirectional quantile information, the information carried by the quantiles of projections. In typi-cal circumstances, they allow for relatively faithful and straightforward retrieval of the directionalquantiles, offering a straightforward probabilistic interpretation in terms of the tangent mass atsmooth boundary points. They can be viewed as a natural, nonparametric extension of “multivari-ate quantiles” yielded by fitted multivariate normal distribution, and, as illustrated on data exam-ples, their construction can be adapted to elaborate frameworks that require more sophisticatedestimation methods than simply evaluating quantiles for empirical distributions. The examples ofsuch frameworks include estimation of extreme quantiles, and directional quantile regression. Theestimates are affine equivariant whenever the estimators of directional quantiles are translationand scale equivariant; quantile regression estimates can be used to obtain bivariate growth charts,with possible adjustments for concomitant variables and other refinements. ReferencesL. Kong and I. Mizera (2008). Quantile tomography: using quantiles with multivariate data.arXiv:0805.0056v1.