I'm learning about Extreme Value Theory. In my class we're discussing the $U(t)$-tail quantile which has the following definition:
For a distribution function $F$, the tail quantile function $U(t)$ is defined such that $$ P(X> U(t) ) = 1 - F(U(t)) = \frac{1}{t}, \,t\geq 1 $$ Suppose that you have a sample $X_1,\ldots, X_n$ from distribution function $F$. Estimation of $U(\frac{1}{p})$ is based on the following result:
For $F\in D(G_\gamma)$, for $\gamma>0$, iff $$ \lim_{t\to\infty}\dfrac{U(tx)}{U(t)} = x^\gamma, $$ $U(tx)\approx U(t)x^\gamma$. Now let $tx = \frac{1}{p}$, and $t = \frac{n}{k}$, where $k$ is a large integer but much smaller than $n$. We then have $$ U\bigl(\frac1p \bigr) \approx U\bigl(\frac{n}{k} \bigr)\bigl( \frac{k}{np}\bigr)^\gamma $$ I think that I understand all this. Now comes the part that I'm having trouble with to grasp:
Let $X_{1,n}, \ldots, X_{n,n}$ be the order statistics of the sample $X_1, \ldots, X_n$. $U(\frac{n}{k})$ can be estimated by the empirical quantile $X_{n-k,n}$. (Why is this a good estimator? Think about the empirical distribution function $\hat{F}(x) = \frac{1}{n}\sum_{i = 1}^n I(X_i\leq x).)$
Question: Why is $X_{n-k,n}$ a good estimator for $U(\frac{n}{k})$?
Intuitively I understand that it might would be a good estimator, since $P\bigl( X > U(\frac1p) \bigr) = p$ so that $U\bigl(\frac1p \bigr) = x_{1 -p}$. Hence you would get that $P\bigl( X > U(\frac{n}{k}) \bigr) = \frac{k}{n}$ and $U(\frac{n}{k}) = x_{1 - k/n}$. A sensible estimator for $x_{1 - k/n}$ would then be $X_{n-k, n}$.
I don't find this satisfying though, especially since I supposedly have to think about the empirical distribution function to understand why this would be a good estimator.