On Kendallʹs Process

LetZ1, …, Znbe a random sample of sizen⩾2 from ad-variate continuous distribution functionH, and letVi, nstand for the proportion of observationsZj,j≠i, such thatZj⩽Zicomponentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution functionKnderived from the (dependent) pseudo-observationsVi, n. This random quantity is a natural nonparametric estimator ofK, the distribution function of the random variableV=H(Z), whose expectation is an affine transformation of the population version of Kendallʹs tau in the cased=2. Since the sample version ofτis related in the same way to the mean ofKn, Genest and Rivest (1993,J. Amer. Statist. Assoc.) suggested that[formula]be referred to as Kendallʹs process. Weak regularity conditions onKandHare 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.