Suppose $X$ is a single observation from a pdf, and I want to test some hypothesis: $H_0:\Theta=\Theta_0$ vs. $H_1:\Theta=\Theta_0^c$.
I want to check whether there exists a UMP level $\alpha$ test for testing these hypotheses.
Now, one of the approaches is definitely to use the concept of MLR, if it exists. But here, I have a doubt.
If I fix $\theta_1<\theta_2$ then the likelihood ratio is INCREASING.
If I fix $\theta_3<\theta_4$ then the likelihood ratio is DECREASING.
Here, $\theta_i$ are all distinct.
Can I still say that the family of pdf has MLR in $X$?