# 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 $p$th sample quantile is asymptotically normal as well. Here is the result:

Theorem. Fix $0 < p < 1$. If $F$ is differentiable at the $p$th 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 $p$th 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.
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