Do we need to have MLR (in same direction) for whole parameter space?

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 $X$?

• What does MLR mean?
– Sycorax
Nov 9 '15 at 17:04

In order for a random variable $t$ with pdf $g(t|\theta)$ to have a MLR, it must be true that for every $\theta_j > \theta_i$, then $\frac{g(t|\theta_j)}{g(t|\theta_i)}$ is either nonincreasing or nondecreasing.
In your case, where $\theta_1<\theta_2$, your LR is increasing, so it is not nonincreasing. In your case, where $\theta_3<\theta_4$, your LR is decreasing, so it is not nondecreasing. Thus, your LR is neither nonincreasing nor nondecreasing for all $\theta_j > \theta_i.$ Since it is neither nonincreasing nor nondecreasing, you cannot say that your family of pdfs has MLR.