I just give an answer as the persons visiting the question shouldnt go through all the conversations. Without imposing another conditions on the parameter \Theta or the types of these two ...

Thank you for reading my book. I am not an expert in Sage, but I can tell you what I had in mind when I wrote that problem. The goal of the exercise is to work with \Theta^6 replicating what is done ...

Saying \Theta' is open is much stronger as saying that it contains some open rectangles. It says that for all \theta \in \Theta', there is an open rectangle R with \theta \in R \subset \Theta' ...

Your MLE for \theta does not seem correct. If \boldsymbol x = (x_1, \ldots, x_n) is an iid sample drawn from a X \sim \operatorname{Pareto}(1,\theta) distribution with density f_X(x) = \theta x^{-\theta-1} \mathbb{1}(x > 1) ...

It will be helpful to start from an explanation of the origin and the proof of the Ramanujan identity. These are hidden (not very deeply) in the theory of elliptic functions. Indeed, Jacobi elliptic ...

I think g(\theta;\theta^0) may have a mistake. If 0 < \theta_2 < 4, then g(\theta;\theta^0)=\frac{1}{4} \int_0^{\theta_2} \ln \frac{1}{\theta_2} dx_{32} = -\frac{\theta_2}{4} \ln \theta_2. ...