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)= Σ i n 1 v, I x1x and F n = n /1 Σ i n 1 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-quantile process.