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

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.