If $Y_1,Y_2,...,Y_n$ ~ $U[0, \theta]$, and we want to test
$$H_0: \theta=\theta_0$$ $$H_1: \theta < \theta_0$$
what would be the likelihood ratio test for a given $\alpha$.
What I know so far:
We know that the MLE of $\theta$ is $Y_\text{MAX}$.
And $F_{Y_\text{MAX}}(y)=P(Y\leq y)=P(\text{all }Y_i \leq y)$
So I think my likelihood function is $(\frac{y}{\theta})^n$
But I don't understand where the supremum comes in and how to find it.
Thanks!