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Approximately unbiased tests based on bootstrap probabilities are consideredfor the exponential family of distributions with unknown expectation parametervector, where the null hypothesis is represented as an arbitrary-shaped regionwith smooth boundaries. This problem has been discussed previously in Efron andTibshirani [Ann. Statist. 26 (1998) 1687-1718], and a corrected p-value withsecond-order asymptotic accuracy is calculated by the two-level bootstrap ofEfron, Halloran and Holmes [Proc. Natl. Acad. Sci. U.S.A. 93 (1996)13429-13434] based on the ABC bias correction of Efron [J. Amer. Statist.Assoc. 82 (1987) 171-185]. Our argument is an extension of their asymptotictheory, where the geometry, such as the signed distance and the curvature ofthe boundary, plays an important role. We give another calculation of thecorrected p-value without finding the ``nearest point'' on the boundary to theobservation, which is required in the two-level bootstrap and is animplementational burden in complicated problems. The key idea is to alter thesample size of the replicated dataset from that of the observed dataset. Thefrequency of the replicates falling in the region is counted for several samplesizes, and then the p-value is calculated by looking at the change in thefrequencies along the changing sample sizes. This is the multiscale bootstrapof Shimodaira [Systematic Biology 51 (2002) 492-508], which is third-orderaccurate for the multivariate normal model. Here we introduce a newly devisedmultistep-multiscale bootstrap, calculating a third-order accurate p-value forthe exponential family of distributions.

Approximately unbiased tests of regions using multistep-multiscale bootstrap resampling