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I'm working through van der Vaart to improve my knowledge on asymptotic statistics, and I'm attempting the following problem.

Let $F_n^{-1}(p)$ be the p-th sample quantile of a sample from a cumulative distribution $F$ on $\mathbb{R}$ that is differentiable with positive derivative at the population p-th-quantile.

Define $F^{-1}(p) = \inf\{x: F(x) \geq p\}$.

Show that $\sqrt(n)(F_n^{-1}(p) - F^{-1}(p))$ is asymptotically normal with mean 0 and variance $\frac{p(1-p)}{f(F^{-1}(p))^2}$

My attempt

To me, this kind of screamed out using the Delta Method (though it's a problem in the M-statistics section of the book). I considered the empirical CDF $\hat{p} = \sum_{i=1}^{n} I\{x_i \leq x\} / n$. By CLT, we get:

$\sqrt{n}(\hat{p} - p) \overset{d}{\rightarrow} N(0, p(1-p))$.

Using the delta method with $F^{-1}(p)$ seems like it'd give the result I need, but I don't see how $F^{-1}(\hat{p})$ is equal to the p-th sample quantile. Am I misunderstanding something here? Is the use of the empirical CDF as the estimator flawed? I feel like I'm generally on the right track, but I can't make the last leap I need to. Any help would be appreciated.

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marked as duplicate by whuber Jan 21 '18 at 22:45

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    $\begingroup$ What is the function $f$ in the denominator? $\endgroup$ – Moss Murderer Jan 21 '18 at 20:27
  • $\begingroup$ The question doesn't specify, but my interpretation was that it was the pdf of the distribution. My thought now is that the use of the empirical distribution as estimator is flawed, and I need to express this as a Z-statistic of some form, but it's not clear at all to me how to do that. $\endgroup$ – Mango Fed Jan 21 '18 at 21:34