From this https://content.wolfram.com/uploads/sites/19/2013/04/Zwilling.pdf :
But I don't know how to solve it .
I know let the derivative of the function L with respect to $\alpha$ and $\beta$ equal to zero , then calculate the $\alpha$ and $\beta$ . But the equation is too complicated .
I wrote the derivative to $\alpha$ and let it equal to 0: $$ \begin{array}{l} \frac{\partial \ln L}{\partial \alpha}=\sum_{i=1}^{n}(\frac{y_{i}}{\alpha}+\frac{1}{\alpha^{2}} \ln (1+\alpha e^{x_{i} \cdot \beta})+(y_{i} + \frac{1}{\alpha})\frac{e^{x_{i} \cdot \beta}}{1+ \alpha e^{x_{i} \cdot \beta}} \\ -\frac{\Gamma^{\prime}(y_{i}+\frac{1}{\alpha})}{\alpha^{2} \Gamma(y_{i}+\frac{1}{\alpha})}+\frac{\Gamma^{\prime}(\frac{1}{\alpha})}{\alpha^{2} \Gamma(\frac{1}{\alpha})} ) = 0 \end{array} $$
But have no idea how to calculate the $\alpha$ along with $\beta$ , and even without of $\beta$ the left equation is complicated , do you have some tricks or technical to solve such equation ?
