Abstract
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 ′ 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 ′ = - n . The new method approximates the bootstrap iteration applied to the bootstrap probability.
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