On the dependence structure of order statistics. (English) Zbl 1065.62087
Summary: 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 \(i<j\), 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 J. W. Tukey [Ann. Math. Statist. 29, 588–592 (1958; Zbl 0086.35601)] and S. H. Kim and H. A. David [J. Statist. Plann. Inference 24, 363–368 (1990; Zbl 0698.62050)]. 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.
This extends earlier results of J. W. Tukey [Ann. Math. Statist. 29, 588–592 (1958; Zbl 0086.35601)] and S. H. Kim and H. A. David [J. Statist. Plann. Inference 24, 363–368 (1990; Zbl 0698.62050)]. 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.
MSC:
| 62G30 | Order statistics; empirical distribution functions |
| 62E10 | Characterization and structure theory of statistical distributions |
| 60E15 | Inequalities; stochastic orderings |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
Keywords:
Concordance ordering; Dispersive ordering; Exponential distribution; Kendall’s tau; Monotone regression dependence; Spearman’s rho; Stochastic increasingnessReferences:
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