The objective is to find an expansion of equation 1 for large values of . In this case a Taylor expansion will not be useful. The solution is found using an integral recurrence relation.
Consider the integral:
(8) |
(9) | |||
(10) |
Now if then
. Therefore, for large Z, this
enables the error function (related to ) to be expanded in terms of
where the absolute values of continue to decrease as increases.
Specifically:
(11) | |||
(12) | |||
(13) |
Equations 2 and 1, and the fact that
can be used to relate and , giving:
(15) |
Therefore by taking the logarithm, and neglecting the term, the
asymptotic expansion for the Z statistic is: