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Suppose my data has 1000 observations of 0, 20 observations of 1000, 10 observations of 10000, and 1 observation of 100000. In my actual data set the numbers are not as nice but I am saying this to give an idea of what my data looks like.

I want to get an estimate of the cumulative distribution function of the mean given the observations. The current plan is to calculate the bootstrap estimate of the mean 1000 times and use the results to generate a distribution of the mean given the observations.

Is this a reasonable procedure? Could somebody help me understand the limitations of such an approach?

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Technically the mean does not have a distribution, but it has a sampling distribution. Otherwise, the bootstrap will be able to provide such an estimate. The empirical quantiles can be highly biased when the sampling distribution is not pivotal. A bias correction can also be applied so that the mean of the ECDF of the sample mean, since they are usually different quantities.

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  • $\begingroup$ Thanks for the clarification on sampling distribution vs distribution. I am not certain what it means for a sampling distribution to be pivotal. I imagine if you cannot tell from the example given then there is no way for me to tell? $\endgroup$ – sendHelpPlease Jun 15 '18 at 13:32

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