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

Abstract

This note displays an interesting phenomenon for the percentiles of independent but non-identical random variables. Let X1,…,Xn be independent random variables obeying non-identical continuous distributions and X(1)≥⋯≥X(n) be the corresponding order statistics. For p∈(0,1), we investigate the 100(1−p)%th percentile X(⌊pn⌋) and prove the 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,…,n and p=12, we show that its median |Med(X1,…,Xn)|=OP(n1∕2⋅(∑k=1nσk−1)−1) as long as {σk}k=1n satisfy certain mild non-dispersion property.