# Sufficient statistic and MLE problem

Let $X_1, \ldots, X_n$ be i.i.d. random variables with pdf $\Theta x^{-2}$, $0 < \Theta \leq x < \infty$.

1. Find a sufficient statistic for $\Theta$.
2. Find the MLE of $\Theta$

-
\begin{align} \hat{\theta}&=\displaystyle \arg\max_\theta L(\theta,X)\\ \hat{\theta}&=\displaystyle \arg\max_\theta \quad \theta^n\, \Pi_i^n X_i^{-2} \end{align} Clearly, you should increase $\hat \theta$ to increase the likelihood. But you can't really do that. Why? (Think about how the relation between $\theta$ and x is defined).