If you have a few data points and you want to calculate confidence intervals of the mean of the data. You know the distribution of the data (e.g. Exponential). Should you in this case use a parametric or a non-parametric bootstrap?
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Can you please define what "a few" points refers at? Also have you might find it more fruitful to turn this question into a hypothesis test (ie. probability that the mean of data is within $[a,b]$). Notice that if you indeed have an exponential you will be looking into an (inverse) scale- rather then a location-shift. Check this thread on [how to compare the mean of two samples whose data fits exponential distributions](https://stats.stackexchange.com/questions/76689) both answers are very strong and I think will be insightful for your task. – usεr11852 Oct 22 '17 at 13:07
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Why use bootstrap at if you know the parametric form? – Michael R. Chernick Oct 22 '17 at 19:04
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Michael, I have only a few data points. I calculate the MLE $\lambda$ from these data points and use the MLE to bootstrap the mean of the exponential distribution. That is the definition of a parametric bootstrap right?(not telling, but asking) – Cardinal Oct 22 '17 at 19:24