On Kendall's 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, such that Z(j) less than or equal to Z(i) componentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution 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 expectation is an affine transformation of the population version of Kendall'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 explicit expression for its limiting covariance function is given. These conditions, which are fairly easy to check, are seen to apply to large classes of multivariate distributions. (C) 1996 Academic Press, Inc.