General Distribution Theory of the Concomitants of Order Statistics

Let $(X_i, Y_i) (i = 1, 2, \cdots, n)$ be $n$ independent $\mathrm{rv}$'s from some bivariate distribution. If $X_{r:n}$ denotes the $r$th ordered $X$-variate, then the $Y$-variate $Y_{\lbrack r:n\rbrack}$ paired with $X_{r:n}$ is termed the concomitant of the $r$th order statistics. The exact and asymptotic distribution theory of $Y_{\lbrack r:n\rbrack}$ and of its rank are studied. The results obtained are applied to a prediction problem in a Round Robin tournament.