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I have a sample of a stochastic process (blue) for which I have developed point estimates for each time step (red). I have then proceeded to bootstrap 95% confidence intervals for the point estimates (black), which is shown in the figure. Now as you can see, the CI doesn't always include the true value of the process. How do I interpret this?

[Stochastic Process (Blue), point estimate line (Red), 95% CI (Black)[1]

Does this mean that my way of finding the CI might be faulty? Or is this a result that can be expected from working with real data?

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  • $\begingroup$ I think you might be confusing your observed values with the true values. If you knew the try population values then you wouldn't need a confidence interval. $\endgroup$
    – dbwilson
    Commented May 12, 2017 at 17:30
  • $\begingroup$ is it me, or the title does not match the body of the question? You say that blue is a sample of the stochastic process in the body, which is not within the confidence interval, but in the title you ask what to do if the true value is not withing the confidence interval. If the blue line is the true value, then it's not a sample. In my opinion, the question has very different meanings depending on what the blue line actually is. $\endgroup$
    – Andrei
    Commented Aug 8, 2022 at 15:20

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I don't see a problem. By definition:

If confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will approximately match the confidence level.

That doesn't mean your confidence interval must contain the true value. In statistics, you never know what the true value is. We don't know with 100% certainty if the true value is inside the interval or not, but we know if we repeat the process 100 times, about 95 of those intervals should bracket the unknown true value.

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  • $\begingroup$ So then, are you suggesting that they should repeat the process 100 times and check if about 95 of those intervals bracket the KNOWN true value? $\endgroup$
    – Andrei
    Commented Aug 8, 2022 at 15:22
  • $\begingroup$ btw, I know that the true value is never known out of samples, and I am aware that the questioner says that blue is a sample, but I think they may have meant to say that it's not a sample, but the actual true value obtained independently and the reason the stochastic process exists is to find estimates cheaper than however the true value was obtained. Of course, I am speculating a bit by choosing which contradictory parts (imo) of the question to pick. $\endgroup$
    – Andrei
    Commented Aug 8, 2022 at 15:26

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