Asymptotic Expansion for a Class of Sample Quantiles

Abstract

Let X_k:n be the kth order statistic (1 ⩽ k ⩽ n) for a random sample of size n from a population with the distribution function F. Let {α _n }, {β_n} (β_n > 0) be sequences of real numbers and let {k_n} (k_n ⩽ n) be a sequence of positive integers. The present article explores the various choices of α _n , β_n and k_n such that under some mild regularity conditions on F, L(Y_n) → n (0,1) as n→∞, where Y_n = (X_kn:n + α _n )⁄β_n. It is further shown that under some additional conditions on F, standard asymptotic expansion (in Edgeworth form) for the distribution of Y_n can be derived.