What are the maximum likelihood estimators of $\eta$ and $\beta$ ($\eta>0$ and $\beta>0$) for an i.i.d. sample of size $n$ from the following density: $f(x_i)=\frac{\beta x_i^{\beta-1} }{\eta ^ {\beta} } \cdot \exp\left(-(x_i/{\eta})^{\beta}\right)$ ?
Ps. According to Forbes et al 2011, p.196, they are the solution to the following equation system, but I don't manage to get the same result...
$$\eta=\left[\frac{\sum_{i=1}^n{x_i^{\beta}}}{n}\right]^{1/\beta}, \beta=\frac{n}{\frac{1}{\eta}\sum_{i=1}^{n}{x_i^{\beta} \cdot \log(x_i)}-\sum_{i=1}^{n}{\log(x_i)}}.$$
Is there an error in the book, or am I making a mistake?