ON KENDALLS PROCESS

Let Z(1), ..., Z(n) be a random sample of size n greater than or equal to 2 from a d-variate continuous distribution function H, and let V-i ,V-n stand for the proportion of observations Z(j), j not equal i, suc h that Z(j) less than or equal to Z(i) componentwise. The purpose of t his paper is to examine the limiting behavior of the empirical distrib ution function K-n derived from the (dependent) pseudo-observations V- i,V-n. This random quantity is a natural nonparametric estimator of K, the distribution function of the random variable V = H(Z), whose expe ctation is an affine transformation of the population version of Kenda ll's tau in the case d = 2. Since the sample version of tau is related in the same way to the mean of K-n, Genest and Rivest (1993, J. Amer. Statist. Assoc.) suggested that root n{K-n(t) - K(t)} be referred to as Kendall's process. Weak regularity conditions on K and H are found under which this centered process is asymptotically Gaussian, and an e xplicit expression for its limiting covariance function is given. Thes e conditions, which are fairly easy to check, are seen to apply to lar ge classes of multivariate distributions. (C) 1996 Academic Press, Inc .