I have a particular PDF with two parameters, specified as:
$$\alpha \beta e^{-\beta x}(1 - e^{-\beta x})^{\alpha - 1}, \alpha > 0, \beta > 0, x_i > 0$$
Given a random iid sample $(x_1, \dots, x_n)$ from the above, I can then write the likelihood function as:
$$L(\alpha, \beta \mid x_1, \dots, x_n) = (\alpha \beta)^n \prod_{i=1}^n e^{-\beta x}(1-e^{-\beta x})^{\alpha-1}$$
Taking the log thereof to make things slightly easier, we then have:
$$\log(L) = n \log(\alpha) + n \log(\beta) - \beta \sum_{i=1}^n x_i + (\alpha-1) \sum_{i=1}^n \log(1 - e^{-\beta x_i})$$
Taking the derivative with respect to $\alpha$ and setting to zero:
$$\frac{\partial}{\partial \alpha} \log(L) = 0 = \frac{n}{\alpha} + \sum_{i=1}^n \log(1-e^{-\beta x_i})$$ $$\hat{\alpha} = \frac{-n}{\sum_{i=1}^n \log(1 - e^{-\beta x_i})}$$
Then, plugging this back into the likelihood and repeating the process for $\beta$, I get something extremely messy and would have to numerically maximize. Did I screw up some math somewhere? No closed form implies that I will need numerical maximization for $\beta$...