Valid Domain

To achieve a relative accuracy of $10^{-3}$ or better, the valid domain is restricted by equations 18 and 33. That is, $Z>4.8$ (or $\log(p) < -14.05$) and $T^6 > 10^3 \frac{15 D^2}{(D+2)(D+4)}$. The former restriction can be expressed in terms of $T$ and $D$ by using equation 31. For large $D$ this becomes:

$$-14.05 > \log(p) \approx -\frac{1}{2} \log(2\pi) - \log(T) - \frac{T^2}{2} + \log(1 - T^{-2} +3 T^{-4})$$ (39)

which is satisfied for $T>4.9$ when $D$ is sufficiently large. For small $D$, the constraint becomes:

$$-14.05 > \log(p) \approx -\frac{D-2}{2} \log(D) - \log(\beta(\frac{D}{2},\frac{1}{2})) - D \log(T) - \frac{D}{D+2} T^{-2}.$$ (40)

Therefore, for very small $D$, large values of $T$ are necessary to satisfy this constraint. Note that both of these latter two equations are more restrictive than the constraint given by 33.

However, these only determine the relative accuracy for each part. But for the overall process the addition of both errors must be within bounds. Therefore, the valid domain was measued empirically (in MATLAB). The exact boundary (where the relative error was $10^{-3}$) is described quite well by equation 31 with $p=-14.05$.

In practice, equation 31 is accurate in all regions where the Z statistic is accurate (that is, where equation 18 holds). Therefore, equation 31 can be used to determine when the domain is valid, by testing whether $\log(p)<-14.5$. Note that a slightly lower threshold is used in practice to be slightly conservative. Outside this region the probability is always never less than $10^{-14}$ was tested empirically and can be confirmed by equations 39 and 40 in the extreme regions.

Furthermore, to speed up the calculation, it is useful to note that if $T \ge 7.5$ and $D \ge 15$, then the domain is valid, whilst for $T<7.5$ and $D \ge 15$ the probability never goes below $10^{-14}$, which allows it to be calculated by conventional methods. Otherwise, for $D<15$ the test involving equation 31 is used.