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I have a time series of velocity values, which is structured into two periods. The first period describes the baseline velocity ("pre"). At the beginning of the second period ("post"), a stimulus is applied and the velocity subsequently changes.

I'd like to assign a confidence interval/significance threshold to the maximum velocity of the "pre" series. I could then determine whether the maximum velocity of the "post" period exceeds that threshold, and thus whether the increase in the "post" period is a response to the stimulus applied, rather than fall within the normal range of maximum velocity to be expected based on the actual maximum velocity and the extent of variance shown in the "pre" period.

I have code that uses stationary bootstrap and the autocorrelation in the velocity time series to generate a bootstrap sampling population of velocity profiles of the "pre" period. I can then calculate a confidence interval of the maximum velocity in this bootstrap population. The problem is that the actual maximum velocity is sampled into most bootstrap samples, and so the variance is small and the resulting confidence interval extremely narrow.

Is there a way to permutate the maximum velocity of the "pre" period based on the overall variance in the "pre" period, and based on this permutation population, calculate a significance threshold? I'm also open to any other suggestions you might have for how to calculate a significance threshold for a "hard" boundary like the maximum.

felix
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    The bootstrap doesn't work for the max, see: [What are examples where a “naive bootstrap” fails?](http://stats.stackexchange.com/a/9722/7290) – gung - Reinstate Monica Aug 14 '15 at 15:29
  • Yes, I know. That's why I'm looking for a different solution now. Thanks for the link though. – felix Aug 14 '15 at 17:12
  • Velocities are *vectors*, so if you're working in more than one dimension, a maximum won't be defined. It sounds like you're working with *speeds*. The distinction, although it might sound pedantic, is huge in this case because it amounts to working with a *univariate* time series instead of the *multivariate* time series suggested by your terminology. – whuber Aug 14 '15 at 17:17
  • That's true. I'm talking about speeds. Thanks for pointing this out! – felix Aug 14 '15 at 17:25

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