Let be i.i.d. random variables with the cumulative distribution function (CDF) . The central limit theorem demonstrates that the sample mean is asymptotically normal (as long as has a finite second moment):
where and are the mean and variance of the random variable with CDF .
It turns out that for any , the th sample quantile is asymptotically normal as well. Here is the result:
Theorem. Fix . If is differentiable at the th quantile with positive derivative , then the sample quantile is asymptotically normal:
This is frequently cited as Corollary 21.5 in Reference 1. A proof of this result can be found in Reference 2 (there, refers to above). (It is essentially an application of the Central Limit Theorem followed by an application of the delta method.)
The numerator of the variance is largest when and gets smaller as we go to more extreme quantiles. This makes intuitive sense: for a fixed level of precision, we need more data to estimate the extreme quantiles as compared to the central quantiles.
The denominator of the variance is largest when the CDF is changing rapidly at the th quantile. This again makes intuitive sense: if is changing a lot there, it means that we have greater probability of getting some samples in the neighborhood of , and so we can estimate it more accurately.
- Van der Vaart, A. W. (2000). Asymptotic statistics.
- Stephens, D. A. Asymptotic distribution of sample quantiles.