On the error term in bahadur's representation of an order statistic

Abstract

Bahadur (1966) presented a representation of an order statistic, giving its asymptotic distribution and the rate of convergence, under weak assumptions on the density function of the parent distribution. In this paper we consider the mean(squared) deviation of the error term in Bahadur’s approximation of the q th sample quantile (qn ). We derive a uniform bound on the mean (squared) deviation of qn , not depending on the value of q. An application of the given result provides the corresponding result for a kernel type estimator of the q th quantile.