ABSTRACT
A multivariate generalization of rank is given by equation (1) using the multivariate definition of sign. We find it more convenient to work with just the summation, and hence will define multivariate rank by
The “/ y ’ is the empirical distribution based on the sample of multivariate X / S . For a given distribution function F, the rank of j t relative to F is given by
(4) Small (1990) alluded to this definition of multivariate rank when he noted that the gradient vector relative to x of EF(\\x — X\\) is a multivariate analog of the univariate rank (which he called quantile). This gradient is R(x; F). The definition in equation (3) is given explicitly in Mottonen and Oja (1995), Chaudhuri (1996), and Koltchinskii (1997). Other notions of multivariate sign and rank, in addition to the obvious coordinatewise one, can be found in Hettmansperger et al. (1992).
