  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 Mathematica. Here we also give a brief explanation of this program.
