AISM 53, 380-403

© 2001 ISM

## 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.

**Source**
( TeX , DVI )