Asymptotic Distribution of Distances Between Order Statistics from Bivariate Populations

Abstract

The exact and limiting distribution of quantiles in the univariate case is well known. Mood [3] investigated the joint distribution of medians in samples from a multivariate population, showing that their distribution is asymptotically multivariate normal. Recently Siddiqui [4] considered the joint distribution of two quantiles and an auxiliary statistic and showed that asymptotically their joint distribution is trivariate normal. Further, he showed the "distances" $X'_{i+l} - X'_i - X'_{i-h}$, ($l$ and $h$ fixed positive integers) between quantiles in the univariate case, when appropriately normalized are asymptotically independently distributed as Chi square r.v.'s with $2l$ and $2h$ d.f. respectively. In this paper the joint distribution of several quantiles from a bivariate population is obtained and it is shown that the distances between quantiles in the separate component populations are independent asymptotically.