# Consistency of sample quantile

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.)

• 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).
– whuber
Commented Mar 12, 2013 at 21:38

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