Calculating the power of permutation tests: A comparison between nonparametric estimators

In this paper we compare nonparametric estimators of the power curve of the one-sample permutation test. These estimators work when $F$ is unknown, as in standard nonparametric frameworks. The asymptotic normal approximation is compared with the conditional expectation of the test and the bootstrap power. Two bootstrap estimators are considered: the classical bootstrap power estimator, derived from the plug-in of the classical empirical distribution function, and the smoothed bootstrap power estimator, obtained from a smoothed estimate of the distribution function. The behavior of these four estimators is compared on the whole power curve by a simulation study which considers five different distribution functions and three different sample sizes. The simulation study shows that the smoothed bootstrap provides the best results in order to estimate high power values for determining the experimental sample size.