The random weighting estimate of quantile process

Abstract

Let X 1, X 2,…, X n be a sample of independent random variables with common distribution function (d f) F and denote by F n the corresponding empirical (d f), where H n is a random weighting estimation of F n ( H n ( x)=∑ i=1 n v i I ( x i x) and F n= 1 n ∑ i=1 nX 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.