Testing regions with nonsmooth boundaries via multiscale bootstrap

A new class of approximately unbiased tests based on bootstrap probabilities is obtained for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region with possibly nonsmooth boundary surfaces such as cones, which appear in, for example, multiple comparisons and hierarchical clustering. The size n′n′ of bootstrap samples is intentionally altered from the size n of the data. A scaling-law of the bootstrap probability leads to our bias corrected p  -values which are calculated by extrapolating the bootstrap probability back to n′=-nn′=-n. The new method approximates the bootstrap iteration applied to the bootstrap probability.