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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 $\Theta$

Any help would be highly appreciated. Please help me!

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1 Answer 1

\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).

Figure this out and you (almost) have your sufficient statistic.

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