There are some subtle but important mathematical typos in the chapter on the Grand Unified Distribution.
While the three table corrections follow from simple algebra, the point about the absolute value is more subtle. To see that problem, let's presume that the motivation for the GUD was adding a Weibull transform the Pearson family. While that may not be exactly true, other derivations lead to similar conclusions. Let $X \sim \text{StdPearson}(a_1, a_2, b_0, b_1, b_2)$, $h(x) = a + sx^{\frac{1}{β}}$ (the Weibull transform), $Y = h(X)$, and $z = \frac{x - a}{s}$ (so that $h^{-1}(x) = z^β$). Then
$$
\begin{align}
\frac{\mathrm{d}}{\mathrm{d}x} \log f_Y(x; a, s, a_1, a_2, b_0, b_1, b_2, β)
&= \frac{\mathrm{d}}{\mathrm{d}x} \log \left( \left| \frac{\mathrm{d}}{\mathrm{d}x} z^β \right| \cdot f_X(z^β) \right)\\
&= \frac{\mathrm{d}}{\mathrm{d}x} \log \left( \left| \frac{β}{s} \cdot z^{β-1} \right| \cdot f_X(z^β) \right)\\
&= \frac{\mathrm{d}}{\mathrm{d}x} \log \left| \frac{β}{s} \cdot z^{β-1} \right| + \frac{\mathrm{d}}{\mathrm{d}x} \log f_X(z^β)\\
&= \frac{β - 1}{s} \cdot \frac{1}{z} - \frac{β}{s} \cdot z^{β-1} \cdot \frac{1}{z^b} \cdot \frac{a_1 z^β + a_2 z^{2β}}{b_0 + b_1 z^β + b_2 z^{2β}}\\
&= -\frac{β}{s} \cdot \frac{1}{z} \cdot \frac{-\frac{β-1}{β} b_0 + \left(a_1 - \frac{β-1}{β} b_1\right) \cdot z^β + \left(a_2 - \frac{β-1}{β} b_2\right) \cdot z^{2β}}{b_0 + b_1 z^β + b_2 z^{2β}}\\
&= \text{GUD}^{(2)}\left(x; a, s, -\frac{β-1}{β} b_0, a_1 - \frac{β-1}{β} b_1, a_2 - \frac{β-1}{β} b_2, b_1, b_2, β\right)\ .
\end{align}
$$
The key is in the third line: $\frac{\mathrm{d}}{\mathrm{d}u} \log |u| = \frac{1}{u}$, which eliminates the absolute-value function. In more general terms, the absolute value function that comes with a Jacobian correction disappears once we start working with the derivative of the log density. Note also that while the $\mathbf{b}$ parameter vector carries directly over to the resulting $\text{GUD}$ distribution despite the transformation, the new $\mathbf{a}$ parameter vector is adjusted relative to the standard Pearson equivalent (with $a_0^{(\text{Pearson})} \triangleq 0$): $\mathbf{a}^{(\text{GUD})} = \mathbf{a}^{(\text{Pearson})} - \frac{β-1}{β} \mathbf{b}^{(\text{Pearson})}$.
While it is, of course, formally possible just to flip the signs of $\mathbf{a}$ where necessary to accommodate the absolute value in the current version of Equation 20.1, it does not seem like this would be desirable or intended. Lines of a table like Table 20.1 would need to be split into two cases, with the sign of $\mathbf{a}$ dependent on the sign of $\frac{β}{s}$. The easiest example demonstrating the problem is the generalised Halphen. While the table would be correct (other than the $\frac{1}{β}$ typo noted above) for the proposed version of Equation 20.1 (without the absolute value), the current version Equation 20.1 (with the absolute value) requires split cases. The generalised Sichel distribution would require an analogous case split.
The proposed correction for the reciprocal inverse Gaussian in the table also assumes that the absolute value will be dropped from Equation 20.1. If there are reasons that the absolute value needs to be maintained, $a_1$ is still wrong but needs to be $+\frac{1}{2}$ instead.
There are some subtle but important mathematical typos in the chapter on the Grand Unified Distribution.
While the three table corrections follow from simple algebra, the point about the absolute value is more subtle. To see that problem, let's presume that the motivation for the GUD was adding a Weibull transform the Pearson family. While that may not be exactly true, other derivations lead to similar conclusions. Let$X \sim \text{StdPearson}(a_1, a_2, b_0, b_1, b_2)$ , $h(x) = a + sx^{\frac{1}{β}}$ (the Weibull transform), $Y = h(X)$ , and $z = \frac{x - a}{s}$ (so that $h^{-1}(x) = z^β$ ). Then
The key is in the third line:$\frac{\mathrm{d}}{\mathrm{d}u} \log |u| = \frac{1}{u}$ , which eliminates the absolute-value function. In more general terms, the absolute value function that comes with a Jacobian correction disappears once we start working with the derivative of the log density. Note also that while the $\mathbf{b}$ parameter vector carries directly over to the resulting $\text{GUD}$ distribution despite the transformation, the new $\mathbf{a}$ parameter vector is adjusted relative to the standard Pearson equivalent (with $a_0^{(\text{Pearson})} \triangleq 0$ ): $\mathbf{a}^{(\text{GUD})} = \mathbf{a}^{(\text{Pearson})} - \frac{β-1}{β} \mathbf{b}^{(\text{Pearson})}$ .
While it is, of course, formally possible just to flip the signs of$\mathbf{a}$ where necessary to accommodate the absolute value in the current version of Equation 20.1, it does not seem like this would be desirable or intended. Lines of a table like Table 20.1 would need to be split into two cases, with the sign of $\mathbf{a}$ dependent on the sign of $\frac{β}{s}$ . The easiest example demonstrating the problem is the generalised Halphen. While the table would be correct (other than the $\frac{1}{β}$ typo noted above) for the proposed version of Equation 20.1 (without the absolute value), the current version Equation 20.1 (with the absolute value) requires split cases. The generalised Sichel distribution would require an analogous case split.
The proposed correction for the reciprocal inverse Gaussian in the table also assumes that the absolute value will be dropped from Equation 20.1. If there are reasons that the absolute value needs to be maintained,$a_1$ is still wrong but needs to be $+\frac{1}{2}$ instead.