The technical details of the new bootstrap method of Shimodaira (2004)
are given here in mathematical proofs as well as a supporting computer
program. Approximately unbiased tests based on the bootstrap
probabilities are considered in Shimodaira (2004) for the exponential
family of distributions with unknown expectation parameter vector, where
the null hypothesis is represented as an arbitrary-shaped region with
smooth boundaries. It has been described in the lemmas of Shimodaira
(2004) that the newly developed three-step multiscale bootstrap method
calculates an asymptotically third-order accurate $p$-value. All the
mathematical proofs of these lemmas are shown here. The straightforward,
though very tedious, calculations involving tensor notations are
verified in Shimodaira (2003), which is, in fact, a computer program for
{\it Mathematica}. Here we also give a brief explanation of this program.