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

Question

I am working on the following exercise:

We are given an observation of a discrete RV $X$ with PMF $f(x \mid \theta)$ and $\theta \in \{0,1,2\}$ as in the table below. Find the LR test for the hypothesis $H_0: \theta = 0$ and list all possible critical areas for such an LR test. From there take a level $\alpha$-test for $\alpha = 0.15$ and find its power function $\beta(\theta)$.

$$\begin{pmatrix} x &f(x \mid 0) &f(x \mid 1) &f(x \mid 2) \\ 1 &3/4 &1/4 &1/3 \\ 2 &1/8 &1/8 &1/3 \\ 3 &1/8 &1/2 &1/6 \\ 4 &0 &1/8 &1/6 \\ \end{pmatrix}$$

I am new to LR tests and can not quite see through its definition yet. Here is what I got so far: We defined the test statistic for the LR test as

$$\lambda(X) := \frac{L(\hat{\theta_0} \mid X)}{L(\hat{\theta} \ \mid X)},$$

where $L$ is the likelihood function. If I am not mistaken we should have:

$$\lambda(X) = \begin{cases} \frac{3/4}{3/4} = 1, & \text{for } X = 1 \\ \frac{1/8}{1/3} = 3/8, & \text{for } X = 2\\ \frac{1/8}{1/2} = 1/4, & \text{for } X = 3\\ \frac{0}{1/6} = 0, & \text{for } X = 4. \end{cases} $$

For the LR tests we defined the critical area $K$ as $K := \{x \in \mathcal{X} \ \mid \lambda(x) < k\}$, where $\mathcal{X}$ is the sample space. In the lecture we then said that we need to find some $k$ such that $\sup_{\theta \in \Theta_0} P_{\theta}(\lambda < k) \le \alpha$, however, I do not see how I should do this here. Could you please help me?

Answer 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.