The random weighting estimate of quantile process

Let X-1, X-2, . . . , X-n be a sample of independent random variables with common distribution function (df) F and denote by F-n the corresponding empirical (df), where H-n is a random weighting estimation of F-n (H-n(x) = Sigma(i=1)(n) nu(i)I((xix)) and F-n = (1)/(n) Sigma(i=1)(n) X-i). In this paper, the random weighting method is applied to the sample q-quantile process. First, the consistency of the random weighting approximation is proved for the distribution of n(1/2) {F-n(-1)(q) - F-1(q)}, and its convergence rate is researched. Second, the weak convergence of a smoothed random weighting estimates error is proved for the sample q-quartile process. (C) 2003 Elsevier Inc. All rights reserved.