Groupe d’études et de recherche en analyse des décisions

Dependence orderings for order statistics and records

Subhash C. Kochar Monash University, Australie

Given a random sample from a continuous distribution, it is observed that the copula linking any pair of order statistics is independent of the parent distribution. To compare the degree of dependence between two such pairs of order statistics, the notion of greater monotone regression dependence, or more stochastically increasing, ordering is considered. Using this notion, it is proved that for \(i\) < \(j\), the dependence of the \(j\)-th order statistic on the \(i\)-th order statistic decreases as i and j draw apart. This extends earlier results of Tukey [Ann.Math.Statist., 1958] and Kim and David [J. Statist.Plan. Inf., 1990]. 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. Similar results for records. It is seen that this problem is intrinsically related to dispersion among spacings.