# Estimates for order statistics in terms of quantiles

## TL;DR

This paper establishes a relationship between the median of the k-th order statistic of independent non-negative variables and the quantile of an averaged distribution, under mild conditions.

## Contribution

It provides a novel equivalence between the median of order statistics and specific quantiles of an averaged distribution for independent non-negative variables.

## Key findings

- Median of k-th order statistic approximates the quantile of order (k-1/2)/n.
- Results hold under mild distributional conditions.
- Applicable to diverse non-negative random variables.

## Abstract

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