Distribution of quantiles in samples from a bivariate population

Let F (x,y) be t he joint d istri b ut ion function of ( X, y), possessing a probability de nsity funct ion f(x,y). Let F I(X) and l>'2(Y) be the marginal distribution fun ctions of X and Y respectively. Let a be a quantile of F I (x) and f3 be a quantile of F2(y). A random sampl e ( X k, Y k), k = 1, 2, . . . , n, is drawn a nd t he values on each variate are ordered so that X: < X; a nd Y;< Y; if i < j. Let i and j be the greatest integers s uch t hat i /n::;' FI (a) and j /n::;' F2(f3), a nd let NI be the number of ele ments (X, Y) such t hat X<X; am! Y < Y. The joint distrib utio n of ( M ,X;, Y;) is obtained and is shown to I'e asymptot icall y ;lOrlllal. Estimates a nd co nfi dence limits on t he parameters of interest are also given.