The Annals of Statistics

Distribution and Expected Value of the Rank of a Concomitant of an Order Statistic

H. A. David, M. J. O'Connell, and S. S. Yang

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Let $(X_i, Y_i)$ be $n$ independent rv's having a common bivariate distribution. When the $X_i$ are arranged in nondecreasing order as the order statistics $X_{r:n} (r = 1,2,\cdots, n)$, the $Y$-variate $Y_{\lbrack r:n\rbrack}$ paired with $X_{r:n}$ is termed the concomitant of the $r$th order statistic. The small-sample theory of the distribution and expected value of the rank $R_{r:n}$ of $Y_{\lbrack r:n\rbrack}$ is studied. In the special case of bivariate normality an illustrative table of the probability distribution of $R_{r,n}$ is given. A more extensive table of $E(R_{r,n})$ is also provided and it is found that asymptotic results require comparatively small finite-sample corrections even for modest values of $n$. Some applications are briefly indicated.

Article information

Ann. Statist., Volume 5, Number 1 (1977), 216-223.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 62F07: Ranking and selection

Order statistics concomitants ranking selection bivariate normal tables


David, H. A.; O'Connell, M. J.; Yang, S. S. Distribution and Expected Value of the Rank of a Concomitant of an Order Statistic. Ann. Statist. 5 (1977), no. 1, 216--223. doi:10.1214/aos/1176343756.

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