Non-asymptotic bounds for percentiles of independent non-identical random variables
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This note displays an interesting phenomenon for percentiles of independent but non-identical random variables. Let X_1,...,X_n be independent random variables obeying non-identical continuous distributions and X^(1)≥...≥ X^(n) be the corresponding order statistics. For any p∈(0,1), we investigate the 100(1-p) 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 X_k∼N(0,σ_k^2) for k=1,...,n and p=1/2, we show that its median | Med(X_1,...,X_n)|= O_P(n^1/2·(∑_k=1^nσ_k^-1)^-1) as long as {σ_k}_k=1^n satisfy certain mild non-dispersion property.
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