TY - JOUR
T1 - On Kendall's process
AU - Barbe, Philippe
AU - Genest, Christian
AU - Ghoudi, Kilani
AU - Rémillard, Bruno
N1 - Funding Information:
Research funds in partial support of this work were granted by the Natural Sciences and Engineering Research Council of Canada, the Fonds pour la formation de chercheurs et l’aide a la recherche du Gouvernement du Quebec, and the Fonds institutionnel de la recherche de l’Universite du Quebec a Trois-Rivieres.
PY - 1996/8
Y1 - 1996/8
N2 - 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.
AB - 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.
KW - Asymptotic calculations
KW - Copulas
KW - Dependent observations
KW - Empirical processes
KW - Vapnik-Červonenkis classes
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U2 - 10.1006/jmva.1996.0048
DO - 10.1006/jmva.1996.0048
M3 - Article
AN - SCOPUS:0030211208
SN - 0047-259X
VL - 58
SP - 197
EP - 229
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
IS - 2
ER -