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Since the likelihood function of $(X_1,\ldots,X_n)$ is

$$L(\theta)=\left(\frac{\theta}{1-\theta^2}\right)^n I(\theta < x_{(1)}) I(\theta < 1/x_{(n)})$$

So the MLE of $\theta$ is $$\hat{\theta}=\min\{x_{(1)},1/x_{(n)}\}$$

But I don't know if $\hat{\theta}$ is strongly consistent estimator. Can anyone tell me?

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  • $\begingroup$ Show that $\hat\theta$ is a monotone function of $n$ since both $x_{(1)}$ and $1/x_{(n)}$ decrease a.s. when $n$ increases. $\endgroup$ – NCh May 8 at 5:25

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