Asymptotic Distribution of Sample Quantiles for Exchangeable Random Variables

Abstract

Summary

The purpose of this article is to investigate the ‘large sample’ properties of sample quantiles when we assume that the basic random variables are exchangeable (ref. Loève (1960) p. 365). It is shown that under different conditions (to be specified below) on the nature of these exchangeable random variables the distribution of the sample quantile X_{r : n} where X_{r : n} is the rth order statistic for the first n exchangeable random variables $\left(\underset{n\to \infty }{\lim }\frac{r}{n}=\hspace{0.17em}p;\hspace{0.17em}0\hspace{0.17em}<\hspace{0.17em}p\hspace{0.17em}<\hspace{0.17em}1\right)$ tends, as n → ∞, to different nondegenerate forms. As an example, the special case of random variables with equicor-related multivariate normal distribution is discussed.