Here's my idea for a bootstrapping method for testing hypotheses about one parameter. Please tell me if you have seen this somewhere before. If not, I'd appreciate pointers for direction of further research.
Suppose I want to do a two-tailed hypothesis test for a population mean ($H_0: \mu=\mu_0, H_a: \mu\ne\mu_0$), and then take a random sample of size $n$ from my population, and get sample mean $M$. A typical bootstrapping method in this case is to shift the sample so that the sample mean equals $\mu_0$, and then sample with replacement many times. The shifting seems to me contrary to the usual spirit of bootstrapping methods: test for the variability of the sample statistic by assuming that your sample is representational. Of course, one can't assume both that the sample is representational and that the null hypothesis is true. But shifting seems to do away with the idea of representation altogether.
So here's an idea: take the original sample and reflect it along the $x=\mu_0$ line, and assume that the resulting set of size $2n$ is representative of the population. Now sample with replacement many samples of size $n$. This way, the individuals in the original sample are assumed to be fairly representative of the population, without supposing an individual even more extreme from the hypothesized mean.
Similar methods could work for other statistics (median, correlation coefficient).
Has anyone come across this idea before?