Asymptotic normality of sample quantiles

Let X_1, \dots, X_n be i.i.d. random variables with the cumulative distribution function (CDF) F. The central limit theorem demonstrates that the sample mean \overline{X}_n is asymptotically normal (as long as F has a finite second moment):

\begin{aligned} \sqrt{n} (\overline{X}_n - \mu) \stackrel{d}{\rightarrow} \mathcal{N}(0, \sigma^2), \end{aligned}

where \mu and \sigma^2 are the mean and variance of the random variable with CDF F.

It turns out that for any 0 < p < 1, the pth sample quantile is asymptotically normal as well. Here is the result:

Theorem. Fix 0 < p < 1. If F is differentiable at the pth quantile F^{-1}(p) with positive derivative f(F^{-1}(p)), then the sample quantile F_n^{-1}(p) is asymptotically normal:

\begin{aligned} \sqrt{n}\left( F_n^{-1}(p) - F^{-1}(p) \right) \stackrel{d}{\rightarrow} \mathcal{N} \left( 0, \dfrac{p(1-p)}{f^2 (F^{-1}(p))} \right). \end{aligned}

This is frequently cited as Corollary 21.5 in Reference 1. A proof of this result can be found in Reference 2 (there, x refers to F^{-1}(p) above). (It is essentially an application of the Central Limit Theorem followed by an application of the delta method.)

The numerator of the variance p(1-p) is largest when p = 1/2 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 f^2 (F^{-1}(p)) is largest when the CDF F is changing rapidly at the pth quantile. This again makes intuitive sense: if F is changing a lot there, it means that we have greater probability of getting some samples in the neighborhood of F^{-1}(p), and so we can estimate it more accurately.

References:

  1. Van der Vaart, A. W. (2000). Asymptotic statistics.
  2. Stephens, D. A. Asymptotic distribution of sample quantiles.