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The quick and to-the-point question I have is:

Can you perform bootstrap resampling on a sampling distribution, using the sampling distribution as if it were an original sample of observations? What are the negative implications?

Here's the long version:

I have a sample of, for example, N=500 non-overlapping weekly return observations of a given mutual fund.

Each return observation represents the return as a percentage of the principal invested at the beginning of a one week period (i.e. a return observation of 1.0 represents zero gain, zero loss; a return observation of 1.05 represents a 5% gain over the week; an observation of 0.92 represents an 8% loss over the week; etc.).

Given my weekly return observations, I would like to compute the sampling distribution of the 1-year mean return.

This is my proposed solution, but I don't know if it's valid:

Given an initial sample of N=500 non-overlapping weekly return observations:

  1. Construct, for example, 1,000,000 bootstrap samples of my original weekly return sample. This step results in 1,000,000 bootstrap samples, each consisting of N=500 randomly selected weekly return observations.
  2. For each bootstrap sample constructed in (1), compute the product of all the weekly return observations in each bootstrap sample, and then annualize the product (e.g. if the product of the 500 weekly return observations was 5.0, then the annualized value would be 5.0 ^ (52/500) = 1.182205...); this yields a sampling distribution of 1,000,000 simulated 1-year return "observations".
  3. Treating the sampling distribution of simulated 1-year return observations from step (2) as a sample of 1-year return observations, construct, for example, 10,000 bootstrap samples of the 1-year return sample. This step results in 10,000 bootstrap samples, each consisting of 1,000,000 randomly selected simulated 1-year return observations.
  4. For each bootstrap sample constructed in (3), compute the mean of the 1-year simulated return observations in each bootstrap sample; this yields a sampling distribution of the mean 1-year simulated return.

The big question in my mind is whether or not step (3) is valid or not. Can I treat the sampling distribution computed in step (2) as a sample from which to perform a second bootstrapping process?

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  • $\begingroup$ By "sampling distribution of the one-year mean return" do you just mean "distribution of the one-year mean return"? If not, please specify what sampling process you have in mind. But before you do that, please check whether your series exhibits serial correlation (which it very likely does), because if so, you will need to do a (semi-)parametric bootstrap that accounts for this. $\endgroup$ – whuber Feb 10 '15 at 21:23
  • $\begingroup$ @whuber, I intended to mean "sampling distribution of the one-year mean return", but I may have used the phrase incorrectly. I would like to resample the sampling distribution of the simulated 1-year return "observations" that I constructed in step (2), as if it were just a regular sample. I'm fairly ignorant of statistics in general, so I don't know if that is "allowed" or not. $\endgroup$ – David Feb 10 '15 at 21:49
  • $\begingroup$ It might be better, then, to ask your question in terms of what you need to accomplish rather than in terms of statistical procedures you might be thinking of but don't fully understand. What is your objective? To estimate a typical one-year return and assess the uncertainty of that estimate? Something else? $\endgroup$ – whuber Feb 10 '15 at 22:11
  • $\begingroup$ @whuber, ultimately I want to know which mutual fund I should invest in. The most straightforward way I can think of to answer that question is to compare the sampling distribution of the mean 1-year return of a given mutual fund with the sampling distribution of the mean 1-year return of every other mutual fund. Of the sampling distributions that don't overlap, the ones situated closest to +infinity are "best". That's my thinking anyway. Again, I could be wrong. $\endgroup$ – David Feb 10 '15 at 22:19
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    $\begingroup$ This is not a statistical comment, but one based on a quarter century of tracking and investing in mutual funds: of all the ways you could use to select them, put this near the bottom. If you want a sense of typical one-year returns and their variability, you could plot the rolling one-year returns (all 449 of them)--but even then you are assuming the future will continue to be much the way things were as long as 500/52 = 10 years ago, which certainly is false--as all the disclaimers on all the prospectuses cheerfully remind you. $\endgroup$ – whuber Feb 10 '15 at 22:26

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