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