Suppose we have only one observation x from X, a discrete random variable, whose distribution depends on a parameter θ ∈ $Θ$ = {$θ1$, $θ2$, $θ3$} and is described by the following table:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| p(x; $\theta 1$) | 0.05 | 0.1 | 0.6 | 0.02 | 0.23 |
| p(x; $\theta 2$) | 0.45 | 0.01 | 0.19 | 0.05 | 0.3 |
| p(x; $\theta 3$) | 0.15 | 0.4 | 0.05 | 0.3 | 0.1 |
- Use the Newman-Pearson lemma to find the most powerful statistical test for testing H0 : θ = θ1 versus H1 : θ = θ2 at a fixed value $\alpha$= 0.05
I know that Neyman-Pearson starts from the ratio of the likelihood, since we have just one observation we start from the ratio of the probabilities under H0 and under H1 so I constructed a table by making $\cfrac{p(x;\theta1)}{p(x;\theta2)}$ I know that we reject when the ratio is below a certain threshold, my question is how can I determine this rejection region?
thanks in advance for your help
There are only three possible critical regions to consider, and they are $C_1=\{1\}, C_2=\{4\},$ and $C_3=\emptyset$. Any other subset $E$ of $\{1,2,3,4,5\}$ will ultimately satisfy $\mathbb{P}(E|\theta =\theta_1)>0.05$. Moreover, $$\mathbb{P}(C_1|\theta=\theta_2)=0.45$$ $$\mathbb{P}(C_2|\theta=\theta_2)=0.05$$ $$\mathbb{P}(C_3|\theta=\theta_2)=0$$ Our desired critical region is $C_1$ i.e. $$\text{Reject } H_0 \iff X=1$$ is the most powerful size test.