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

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  • Xia, Dong

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.

Suggested Citation

  • Xia, Dong, 2019. "Non-asymptotic bounds for percentiles of independent non-identical random variables," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 111-120.
    • Handle: RePEc:eee:stapro:v:152:y:2019:i:c:p:111-120
    DOI: 10.1016/j.spl.2019.04.018
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    References listed on IDEAS

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    1. Iain M. Johnstone & Bernard W. Silverman, 1997. "Wavelet Threshold Estimators for Data with Correlated Noise," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 319-351.
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