ON AFFINE EQUIVARIANT MULTIVARIATE QUANTILES

BIMAN CHAKRABORTY

Department of Statistics and Applied Probability, The National University of Singapore, 10, Kent Ridge Crescent, Singapore 119260, Singapore

(Received August 3, 1998; revised June 14, 1999)

Abstract. An extension of univariate quantiles in the multivariate set-up has been proposed and studied. The proposed approach is affine equivariant, and it is based on an adaptive transformation retransformation procedure. Bahadur type linear representations of the proposed quantiles are established and consequently asymptotic distributions are also derived. 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.

Key words and phrases: Bahadur representation, L-estimates, multivariate ranks, Q-Q plots, quantile contour plots, transformation-retransformation.