# Non-asymptotic bounds for percentiles of independent non-identical random variables

arXiv: https://arxiv.org/abs/1808.07997 · HTML: https://mlsanity.com/paper/1808.07997/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Aug 2018.

## Built on by (most cited first, 3 of 3)

- [Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians](https://mlsanity.com/paper/2301.09024/) (2023)
- [Entangled Mean Estimation in High-Dimensions](https://mlsanity.com/paper/2501.05425/) (2025)
- [Performance of the empirical median for location estimation in heteroscedastic settings](https://mlsanity.com/paper/2501.16956/) (2025)

## Abstract (arXiv)

This note displays an interesting phenomenon for percentiles of independent but non-identical random variables. Let X1,cdots,Xn be independent random variables obeying non-identical continuous distributions and X(1)≥ cdots≥ X(n) be the corresponding order statistics. For any p∈(0,1), we investigate the 100(1-p)%-th percentile X(pn) and prove non-asymptotic bounds for X(pn). In particular, for a wide class of distributions, we discover an intriguing connection between their median and the harmonic mean of the associated standard deviations. For example, if Xk~N(0,σk2) for k=1,cdots,n and p=frac12, we show that its median big|rm Medbig(X1,cdots,Xnbig)big|= OPBig(n1/2·big(sumk=1nσk-1big)-1Big) as long as σkk=1n satisfy certain mild non-dispersion property.