On Kendall's process

Abstract

Let Z1 , ..., Zn be a random sample of size n ≥ 2 from a d-variate continuous distribution function H, and let Vi, n, stand for the proportion of observations Zj, j ≠ i, such that Zj ≤ Zi componentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution function Kn derived from the (dependent) pseudo-observations Vi, 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 τ is related in the same way to the mean of Kn, Genest and Rivest (1993, J. Amer. Statist. Assoc.) suggested that √n{Kn(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. © 1996 Academic Press, Inc.