I have 100 data points from a random process. How would I go about placing a confidence interval around the estimate of $\Pr(X>x)$? The distribution function is unknown and positively skewed. My first inclination would be to use a bootstrap based on the material I have read for this class, but is there some other way to do this?

**Answer**

Yes, there are other types of confidence intervals (CI). One of the most popular CI are based on the *Dvoretzky–Kiefer–Wolfowitz inequality*, which states that

$$P\left[\sup_{x}\vert \hat{F}_n(x)-F(x)\vert>\epsilon\right]\leq 2\exp(-2n\epsilon^2).$$

Then, if you want to construct an interval of level $\alpha$ you just have to equate $\alpha=2\exp(-2n\epsilon^2)$, which leads to $\epsilon = \sqrt{\dfrac{1}{2n}\log\left(\dfrac{2}{\alpha}\right)}$. Consequently, a confidence band for $F(x)$ is $L(x)=\max\{\hat{F}_n(x)-\epsilon,0\}$ and $U(x)=\min\{\hat{F}_n(x)+\epsilon,1\}$. You may want to work out the details and adapt this to $P[X>x]=1-F(x)$ (since you tagged this as self-study).

This presentation provides other details that might be of interest.

**Attribution***Source : Link , Question Author : Eric Brady , Answer Author : Rodrigo Zepeda*