On the dependence structure of order statistics

Authors:
Jean Avérous;Christian Genest;Subhash C. Kochar
Affiliations:
Laboratoire de statistique et de probabilités, Université Paul-Sabatier, 118, route de Narbonne, Toulouse 31062, France;Département de mathématiques et de statistique, Université Laval, Québec, Canada G1K 7P4;Indian Statistical Institute, 7 SJS Sansanwal Marg, New Delhi 110016, India
Venue:
Journal of Multivariate Analysis
Year:
2005

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Abstract

Given a random sample from a continuous variable, it is observed that the copula linking any pair of order statistics is independent of the parent distribution. To compare the degree of association between two such pairs of ordered random variables, a notion of relative monotone regression dependence (or stochastic increasingness) is considered. Using this concept, it is proved that for i , the dependence of the jth order statistic on the ith order statistic decreases as i and j draw apart. This extends earlier results of Tukey (Ann. Math. Statist. 29 (1958) 588) and Kim and David (J. Statist. Plann. Inference 24 (1990) 363). The effect of the sample size on this type of dependence is also investigated, and an explicit expression is given for the population value of Kendall's coefficient of concordance between two arbitrary order statistics of a random sample.