Estimates for order statistics in terms of quantiles

Abstract

Let $X_1,\dots ,X_n$ be independent non-negative random variables with cumulative distribution functions $F_1,F_2,\dots ,F_n$, each satisfying certain (rather mild) conditions. We show that the median of $k$-th smallest order statistic of the vector $(X_1,\dots ,X_n)$ is equivalent to the quantile of order $(k-1/2)/n$ with respect to the averaged distribution $F=\frac{1}{n}{\sum }_{i=1}^n{F}_i$.

A small observation published in Zapiski Nauchnykh Seminarov POMI, 457 (2017), 265--275. arXiv admin note: substantial text overlap with arXiv:1609.02126