The Annals of Probability

On the Average Difference Between Concomitants and Order Statistics

Prem K. Goel and Peter Hall

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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 $x$.

Article information

Ann. Probab., Volume 22, Number 1 (1994), 126-144.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


Goel, Prem K.; Hall, Peter. On the Average Difference Between Concomitants and Order Statistics. Ann. Probab. 22 (1994), no. 1, 126--144. doi:10.1214/aop/1176988851.

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