On the Average Difference Between Concomitants and Order Statistics

Abstract

For a sequence of bivariate pairs $(X_i, Y_i)$, the concomitant $Y_{\lbrack i\rbrack}$ 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 $Y_{\lbrack i\rbrack} - Y_{(i)}$. 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 $x$.