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:
- 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.
- 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".
- 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.
- 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?