I'm new to statistics and recently I learnt about non-parametric estimation. On the estimation of empirical quantiles, many notes talk about the asymptotic behavior of sample quantiles. The proof is based on the consistency of the empirical quantile estimator, then applying Slutsky's theorem in combination with the delta method (assuming the cdf is $C^{1}$ at the considered point).

However, it seems to give a very old and trivial result stating that the empirical quantile converges in probability to the true value (with $C^1$ cdf hypothesis). I've asked my professor and he just give the suggestion using Hoeffding's inequality (he doesn't want to spend his time explaining such an easy problem). So, could someone please provide me the proof of this, or a reference covering this issue (e.g. a note on the internet, or link to a pdf file.)

  • 2
    $\begingroup$ Your assertion is not quite true. Let the CDF be $F$ and the quantile be $q$. If $F(x)=q$, you need to add some condition like $F(x+\varepsilon) - F(x)\ne 0$ for all $\varepsilon\ne 0$: this assures the quantile is unique (and gives some clues about how the proof might proceed). $\endgroup$
    – whuber
    Commented Mar 12, 2013 at 21:38

2 Answers 2


The proof using Hoeffding's inequality can be found at page 8 of these notes.

  • 1
    $\begingroup$ thank you very much, these notes is absolutly what i am searching. I just want to remark that exactly the theorem is found on its 10th page. The remark of whyber is also true. I forget the hypothesis of uniqueness for the quantile. $\endgroup$
    – ctNGUYEN
    Commented Mar 13, 2013 at 8:36
  • $\begingroup$ The link seems to be broken. Is there any other document presenting this proof? $\endgroup$ Commented Nov 20, 2022 at 11:58

Another way, perhaps more indirect, to prove this is to note that the sample quantile is also an M-estimator, specifically when $\rho(y_i,\theta)=\alpha(y_i-\theta)_+ + (1-\alpha)(\theta-y_i)_+$.

Because of that it enjoys all the properties of M-estimators, one of them being that it is consistent in probability (converges in probability) to its mean, which is the real/population quantile.

The (general M-estimator) proof of this theorem uses Slutsky and Taylor-expansion/Delta-method.

Your question doesn't ask to show the rate of convergence, so using Hoeffding is perhaps an overkill.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.