Testing a nonstandard hypothesis: constructing test statistic, finding rejection region and obtaining $p$-value

Question

I have a sample of size $n=1$ (a single observation $x_1$) from a random variable $X\sim N(\mu,\sigma^2)$. The variance $\sigma^2$ is known, but the expectation $\mu$ is unknown. I would like to test the null hypothesis $$H_0\colon \quad \mu=0 \quad \text{or} \quad \mu=2$$ against the alternative $$H_1\colon \quad \mu=1 \quad \text{or} \quad \mu=3.$$ My significance level is $\alpha=0.05$.

Questions: How do I test this using the frequentist approach? Specifically:

1. What test statistic may I use?
2. How do I find the rejection region?
3. How do I calculate the $p$-value?

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_{This is related to my continuing efforts to understand the $p$-value as in}

1. _{"Is the following textbook definition of $p$-value correct?",}
2. _{"Does $p$-value ever depend on the alternative?",}
3. _{"Defining extremeness of test statistic and defining $p$-value for a two-sided test" and}
4. _{"$p$-value: Fisherian vs. contemporary frequentist definitions".}

_{and my studies of likelihood ratio testing:}

5. _{"Asymptotic null distribution of the LR statistic with point null and point alternative"}
6. _{"Failing to obtain $\chi^2(1)$ asymptotic distribution under $H_0$ in a likelihood ratio test"}

Answer 1

Some intuition can be gained from considering a likelihood ratio (LR) test. The following interactive graph shows how the LR varies with $\sigma$. Three snapshots are provided below.

The dotted red line is the likelihood under $H_0$, $\text{L}(x_1\mid \mu=0 \ \text{or} \ \mu=2)$.
The dashed blue line is the likelihood under $H_0 \cup H_1$, $\text{L}(x_1\mid \mu=0 \ \text{or} \ \mu=1 \ \text{or} \ \mu=2 \ \text{or} \ \mu=3)$.
The solid black line is the likelihood ratio, $\text{LR}(x_1\mid\dots) = \frac{ \text{L}(x_1\mid \mu=0 \ \text{or} \ \mu=2) }{ \text{L}(x_1\mid \mu=0 \ \text{or} \ \mu=1 \ \text{or} \ \mu=2 \ \text{or} \ \mu=3) }$.

The top figure is for $\sigma=0.25$. The middle figure is for $\sigma=0.5$. The bottom figure is for $\sigma=1$.

Observations

• For small $\sigma$ we have good precision and the LR test has two rejection regions (RRs): around $1$ and well beyond $2$. When $\sigma$ grows, the RR around $1$ shrinks and eventually disappears (for a fixed significance level $\alpha$). The RR to the right of $2$ starts further and further away.
• The precise boundaries of RRs depend on the significance level in addition to $\sigma$.
  (I know I have specified $\alpha=0.05$, but this is a general comment.)
• The $p$-value is the area under the LR curve where $x$ is such that $\text{LR}(x\mid\dots)\leq\text{LR}(x_1\mid\dots)$ for the particular $x_1$ that we have observed.