most of the questions and topics I found about the MLE on this site focus on concrete examples, where mostly the standard strategy of maximizing via differentiating was the way to go.
My situation is the following: Assume we have a sample $X_1, \ldots, X_n$ generated by distributions, which depend on two Parameters $\theta = (\theta_1, \theta_2) \in {\mathbb{R}^+}^2,$ such that our parameter space is open and not compact. Furthermore assume, that the log-Likelihood function is differentiable regarding the parameters but not in a nice and clean way (so we can't solve explicitely for $\theta_1, \theta_2$).
What is your strategy (theorems at hand etc.) for proving existence or non-existence of the MLE in these "rougher" situations?
Concrete example: Let our $X_i$ be identically and idepently distributed with a CDF $$F_\theta(x) := \begin{cases} 0, & \text{for } x \leq 0 \\ 1- \exp(- \frac{\theta_1}{\theta_2}x^{\theta_2}), & \text{for } 0 < x < 1 \\ 1- \exp(- \frac{\theta_1}{\theta_2} \cdot (x^{\theta_2} - (x-1)^{\theta_2})), & \text{for } 1 \leq x \end{cases}, $$ where $\theta \in {\mathbb{R}^+}^2.$ How to start proving the existence (or non existence) of the MLE?
Kind regards, fixfoxi
If the likelihood isn’t differentiable use momentum gradient descent or another method for finding relative maxima.
If the likelihood is differentiable then use a numerical method for finding zeros such as Newton’s method and check the Hessian is negative definite.