In the sampling version of permutation testing, the one-sided P-value is Pˆ = (X + 1)/(N + 1), where X is the number of permutation test statistics that are as large or larger than the observed test statistic. Suppose the true P-value (for the exhaustive test, conditional on the observed data) is p.
(a) What is the variance of Pˆ?
(b) What is the variance of Pˆ2 for the two-sided test (assuming that p is not close to 0.5, where p is the true one-sided P-value?)