ON AFFINE EQUIVARIANT MULTIVARIATE QUANTILES

An extension of univariate quantiles in the multivariate set-up has been proposed and studied. Unlike coordinatewise quantile vectors and geometric (or spatial) quantiles considered by some earlier authors [see Chaudhuri (1996, JASA), Koltchinskii (1997, Ann. Stat.)], our approach is affine equivariant, and it is based on an adaptive transformation retransformation procedure. As applications of these multivariate quantiles, we develop some affine equivariant quantile contour plots which can be used to study the geometry of the data cloud as well as the underlying probability distribution and to detect outliers. These quantiles can also be used to construct affine invariant versions of multivariate Q-Q plots which are useful in checking how well a given multivariate probability distribution fits the data and for comparing the distributions of two data sets. We illustrate these applications with some simulated and real data sets. We also indicate a way of extending the notion of univariate L-estimates and trimmed means in the multivariate set-up using these affine equivariant quantiles.