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Let $X_1,\dots X_n$ be i.i.d. samples from an unknown distribution. We know the distribution has uniformly bounded probability density function $f(x)$. Let $1>\tau_1>\tau_2>0$ be two quantile levels and $\hat{X}^{\tau_1}$ and $\hat{X}^{\tau_2}$ be the corresponding sample quantile points. Note that the sample quantile is not unique. Now I want to give a sharp bound for the distance \begin{equation} |\hat{X}^{\tau_1}-\hat{X}^{\tau_2}|. \end{equation} This problem also has a special case like the finding distance between two neighboring quantiles (i.e. $\tau_1-\tau_2=1/n$).

For simplicity we may assume $X_i$'s all sampled from standard normal distribution. So my question is, are there any elegant proofs for this problem? I would be grateful if somebody provide any reference about concentration results for median-type estimators.

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  • $\begingroup$ I think you are looking for the results on the distribution of order statistics (en.wikipedia.org/wiki/Order_statistic). $\endgroup$
    – Anthony Quas
    Commented Jan 20, 2020 at 17:57
  • $\begingroup$ @AnthonyQuas Thanks, but I think I need sharper inequalities, like exponential-type inequality or even Bahadur representation, the results like variance is not enough $\endgroup$
    – aurora_borealis
    Commented Jan 21, 2020 at 1:03
  • $\begingroup$ Would an asymptotic bound serve your needs ? $\endgroup$
    – Gilles Mordant
    Commented Jan 23, 2020 at 17:01

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