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

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 \(x\).


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