On the average difference between concomitants and order statistics. (English) Zbl 0793.60019

Goel, Prem K.; Hall, Peter

Ann. Probab. 22, No. 1, 126-144 (1994).

Summary: For a sequence of bivariate pairs \((X_ i,Y_ i)\), the concomitant \(Y_{[i]}\) of the \(i\)-th largest \(x\)-value \(X_{(i)}\) equals that value of \(Y\) paired with \(X_{(i)}\). In assessing the quality of a file- merging or file-matching procedure, the penalty for incorrect matching may often be expressed as the average value of a function of the difference \(Y_{[i]}-Y_{(i)}\). We establish strong laws and central limit theorems for such quantities. Our proof is based on the observation that if \(G_ x(\cdot)\) denotes the distribution function of \(Y\) given \(X=x\), then \(G_ X(Y)\) is stochastically independent of \(X\), even though \(G_ x(\cdot)\) depends numerically on \(x\).

MSC:

60F05	Central limit and other weak theorems
60F15	Strong limit theorems
62G30	Order statistics; empirical distribution functions

Keywords:

bivariate order statistics; central limit theorem; concomitants; file- matching; file-merging; induced order statistics; strong law of large numbers