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We consider hypothesis testing for the null hypothesis being represented asan arbitrary-shaped region in the parameter space. We compute an approximatep-value by counting how many times the null hypothesis holds in bootstrapreplicates. This frequency, known as bootstrap probability, is widely used inevolutionary biology, but often reported as biased in the literature. Based onthe asymptotic theory of bootstrap confidence intervals, there have been somenew attempts for adjusting the bias via bootstrap probability without directaccess to the parameter value. One such an attempt is the double bootstrapwhich adjusts the bias by bootstrapping the bootstrap probability. Another newattempt is the multiscale bootstrap which is similar to the m-out-of-nbootstrap but very unusually extrapolating the bootstrap probability to $m=-n$.In this paper, we employ these two attempts at the same time, and call the newprocedure as multiscale-double bootstrap. By focusing on the multivariatenormal model, we investigate higher-order asymptotics up to fourth-orderaccuracy. Geometry of the region plays important roles in the asymptotictheory. It was known in the literature that the curvature of the boundarysurface of the region determines the bias of bootstrap probability. We foundout that the curvature-of-curvature determines the remaining bias of doublebootstrap. The multiscale bootstrap removes these biases. The multiscale-doublebootstrap is fourth order accurate with coverage probability erring only$O(n^{-2})$, and it is robust against computational error of parameterestimation used for generating bootstrap replicates from the null distribution.

Higher-order accuracy of multiscale-double bootstrap for testing regions