Asymptotic Expansions for Sample Quantiles

Abstract

This paper deals with an Edgeworth-type expansion for the distribution of a sample quantile. As the sample size $n$ increases, these expansions establish a higher order approximation which holds uniformly for all Borel sets. If the underlying distribution function has $s + 2$ left and right derivatives at the true quantile, the error of the approximation is of order $O(n^{-(s+1)})$. From this result asymptotic expansions for the distribution functions of sample quantiles and for percentage points are derived.