Constructing an LR test for a concrete example and find its critical areas and power function

Solution 1:

Assuming a two-sided alternative hypothesis, your work so far is correct.

Here is a hint on how to proceed further:

A critical region of the form $\lambda(x)<k$ essentially means that you reject $H_0$ for small values of $\lambda$.

Observe that

$$\lambda(4)<\lambda(3)<\lambda(2)<\lambda(1) \tag{$\star$}$$

Your critical region $R$ (say) would consist of sample points taken according to $(\star)$. So possible critical regions would be $\{4\}$ or $\{4,3\}$ etc. depending on the level restriction on the test. For a level $\alpha=0.15$ test, $4$ is the first sample point to enter $R$, followed by $3$, and so on until the size of the test exceeds $0.15$.

As you can see,

$$P_{H_0}(X=4)=0<0.15$$

And

$$P_{H_0}(X=4)+P_{H_0}(X=3)=\frac18=0.125<0.15$$

But $$P_{H_0}(X=4)+P_{H_0}(X=3)+P_{H_0}(X=2)=\frac28=0.25>0.15$$

So the tests with critical regions $R=\{4\}$ and $R=\{4,3\}$ are both valid level $0.15$ likelihood ratio tests. Of course, the more points you add in $R$, higher is the power of the test. On the other hand, the test with $R=\{4,3,2\}$ is not a level $0.15$ likelihood ratio test.