Order statistics in overlapping samples, moving order statistics and U-statistics

Abstract

Let X 1 , X 2 , … be a sequence of independent random variables with common distribution function F ( x ). Also let X y:n (i) ( i = 1,2, …) denote the r th order statistic in ( X i …, X i +n−1 ). An expression is obtained for cov ( X r:n1 (1) , X s:n2 (n1 + 1-c) ) in terms of the first two moments of order statistics in a sample of n'=n 1 +n 2 −c drawn from F ( x ). This result is used to study the behaviour of order statistics in moving samples and to unify certain results for linear functions of order statistics expressible as U-statistics.