# Consistency of MLE for uniform distribution $U[\theta,1/\theta]$ where $0<\theta<1$

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?

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