On Kendall's process

Let Z₁ , ..., Z_n be a random sample of size n ≥ 2 from a d-variate continuous distribution function H, and let V_{i, n}, stand for the proportion of observations Z_j, j ≠ i, such that Z_j ≤ 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, 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 K_n, Genest and Rivest (1993, J. Amer. Statist. Assoc.) suggested that √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.