Let $X_1, \ldots , X_n$ be $n$ i.i.d. random variables with density $f_\theta$ with respect to the Lebesgue measure. Find the MLE of $\theta$.

$f_\theta(x)=r\theta^rx^{-(r+1)}; \quad x \ge \theta; \quad \theta>0;\quad r>0-\text{constant}$

What I did:

$$L(X_1,X_2,\ldots,X_n;\theta)=\prod_{i=1}^nr\theta^rX_i^{-(r+1)}=(r\theta^r)^n\prod_{i=1}^nX_i^{-(r+1)}\\ \implies\log L(X_1,X_2,\ldots,X_n;θ)=n\log r+nr \log \theta-(r+1)\sum_{i=1}^n\log X_i,$$then I took derivative with respect to $\theta$:

$nr/\theta=0$; What is wrong with my computation?


$\log \theta $ is monotone increasing function of $\theta$, and as you want the largest possible point in $\Theta$, and using the restriction that $x \ge \theta$, you get $$ x_i \ge x_{(1)} \ge \theta, $$ hence, $$ \hat{\theta}_n = x_{(1)}. $$


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