Likelihood Ratio Test for uniform distribution

Question

I'm doing self-study here, and have a question about the LRT for uniform distribution.

$X_1, X_2,\ldots, X_n$ are i.i.d $\operatorname{Uniform}(0, \theta)$

$H_0: \theta = \theta_0 = 1$
$H_1: \theta < 1$

What I tried so far:

$$\lambda(\theta) = \frac{\prod(X_{(n)} < \theta_0)}{\theta^{-n} \prod(X_{(n)} < \theta)}$$

How to deal with the product of the indicate function here?

Thanks!

Answer 1

The MLE under the alternative hypothesis is $X_{(n)}$.

The MLE under the null hypothesis is $1,$ since that is the only number in the parameter space if the null hypothesis is true.

Therefore the numerator in $\lambda(\theta)$ is $1.$ You have $$ \lambda(\theta) = \begin{cases} \dfrac 1 {\theta^{-n}} = \theta^n & \text{if } X_{(n)} \le \theta\le 1, \\ \\ 0 & \text{otherwise.} \end{cases} $$