The Dvoretzky–Kiefer–Wolfowitz inequality is the following:
$Pr(\text{sup}|\hat{F}_n(x)-F(x)|>\epsilon)\leq 2\exp(-2n\epsilon^2)$,
and it predicts how close an empirically determined distribution function will be to the distribution function from which the empirical samples are drawn. Using this inequality we are able to draw confidence intervals (CI's) around $\hat{F}_n(x)$ (ECDF). But these CI's will be equal in distance around every point of the ECDF .
What I wonder, is there another way to construct a CI around the ECDF?
Reading about ordered statistics we find that the asymptotic distribution of the ordered statistic is the following:
Now, first off, what does the $np$-index with those symbols mean?
Main question: are we able to use this result, together with the delta method (see below), to provide CI's for the ECDF. I mean, the ECDF is a function of the ordered statistic, right? But at the same time the ECDF is a non-parametric function, so is this a dead end?
We know that $E(\hat{F}_n(x))=F(x)$ and $\text{Var}(\hat{F}_n(x))=\frac{F(x)(1-F(x))}{n}$
I hope I'm clear as to what I'm getting at here, and appreciate any help.
EDIT:
Delta method: If you have a sequence of random variables $X_n$ satisfying
and $\theta$ and $\sigma^2$ are finite, then the following is satisfied:
for any function g satisfying the property that $g′(\theta)$ exists, is non-zero valued, and is polynomially bounded with the random variable (quote wikipedia)