Compute the Maximum Likelihood Estimator of a pareto Distribution for statistic...

Question

Can anyone please help compute the Maximum Likelihood Estimator for:

$X_1,...X_n$ iid with $f(x; \theta) = \frac1 \theta(1+x)^{\frac{-\theta +1}\theta}$

For the statistic $T(x) = (X_1,...,X_n)$ (which I guess just means compute the MLE in a general sense for the sample)

I understand how the likelihood function works with this, but I dont understand how I can get the joint distribution of this? I just multiply all the individual distributions together (because they're independent) but I'm still not sure what this looks like.

Any help would be greatly appreciated

Answer 1

\begin{align} \mathcal L(x;θ)&=\prod_{k=1}^nf(x_k;θ)=\frac1{θ^n}\prod_{k=1}^n(1+x_k)^{-1+\frac1θ}\\[0.2cm] \mathcal l(x;θ)=\ln{(\mathcal L(x;θ))}&=-n\ln{θ}+\left(\frac1θ-1\right)\sum_{k=1}^{n}\ln{(1+x_k)}\\ \frac{d}{dθ}\mathcal{l}(x;θ)&=-\frac nθ-\frac{1}{θ^2}\sum_{k=1}^{n}\ln{(1+x_k)}\overset{!}=0 \iff \hat{θ}=-\frac1n\sum_{k=1}^{n}\ln{(1+x_k)}\end{align}