ABSTRACT
Let X 1,…, X n be a sample from a distribution F on the real line and assume that F is continuous so that all observations are distinct with probability 1. We may then arrange the observations in increasing order without ties, X (n:1) < X (n:2) < ⋯ < X (n:n). These variables are called the order statistics. For clarity, we will usually drop the dependence on n in the notation and write simply X (k) = X (n : k) as the kth order statistic and let X (1) < ⋯ < X (n) denote the order statistics. For 0 < p < 1, the pth quantite of F is defined as x p = F −1 (p), and the pth sample quantile is defined as X (k) where k = [np] = the ceiling of np (the smallest integer greater than or equal to np). If the density f(x) exists and is continuous and positive in a neighborhood of some quantiles, then the joint distribution of the corresponding sample quantiles is asymptotically normal. We give the proof for two quantiles; the extension to many quantiles is easy.
