  The general philosophy for bootstrap or permutation methods for testing
hypotheses is to simulate the variation of the test statistic by generating the
sampling distribution which assumes both that the null hypothesis is true, and
that the data in the sample is somehow representative of the population. This
philosophy is inapplicable for testing hypotheses for a single parameter like
the population mean, since the two assumptions are contradictory (e.g., how can
we assume both that the mean of the population is $\mu_0,$ and that the
individuals in the sample with a mean $M \ne \mu_0$ are representative of the
population?). The Mirror Bootstrap resolves that conundrum. The philosophy of
the Mirror Bootstrap method for testing hypotheses regarding one population
parameter is that we assume both that the null hypothesis is true, and that the
individuals in our sample are as representative as they could be without
assuming more extreme cases than observed. For example, the Mirror Bootstrap
method for testing hypotheses of one mean uses a generated symmetric
distribution constructed by reflecting the original sample around the
hypothesized population mean $\mu_0$. Simulations of the performance of the
Mirror Bootstrap for testing hypotheses of one mean show that, while the method
is slightly on the conservative side for very small samples, its validity and
power quickly approach that of the widely used t-test. The philosophy of the
Mirror Bootstrap is sufficiently general to be adapted for testing hypotheses
about other parameters; this exploration is left for future research.
