Using the bootstrap in testing symmetry versus asymmetry

One approach to testing the hypothesis of symmetry versus asymmetry is to use estimates of the center of symmetry in nonparametric tests about a known center. The problem with this approach is that the resulting tests are not distribution-free. Hence, one cannot compute critical values, p-values, or probabilities of errors for these tests. We propose to sidestep this problem with a symmetric bootstrap procedure which uses bootstrap samples from (a smoothed version of) the closest symmetric distribution to the empirical distribution of the data to estimate these values. In this paper, we report simulation experience with this approach in testing the hypothesis of symmetry using the Schuster and Narvarte estimator of the center of symmetry in the so-called Butler sup norm test for symmetry about a known center.