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?

  • $\begingroup$ How to plot this in R with multiple groups on the ecdf $\endgroup$ – Kirti Sehgal Apr 5 at 18:07

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.

  • $\begingroup$ Thansk for this. This inequality is discussed nowhere in the material for my class, so I'm not sure if this is what they are actually looking for. Whether or not this ultimately is the answer they were looking for though this is super useful, and seems like it should lead to a solution to my problem. $\endgroup$ – Eric Brady Apr 8 '13 at 20:06
  • $\begingroup$ I am glad to see you find it interesting. Did you study the asymptotic normallity of the ECDF? $\endgroup$ – Person Apr 8 '13 at 21:19
  • $\begingroup$ No actually. This isn't in any of the material we covered. In this class, we've only studied confidence intervals around estimated parameters and quantiles. I think we are "supposed" to solve this problem using an estimate of population proportion based on the textbook and notes, but I'm still not clear if this is appropriate or not. That's the only reason I hadn't marked this correct yet. $\endgroup$ – Eric Brady Apr 9 '13 at 14:28

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