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?

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:

  1. "Asymptotic null distribution of the LR statistic with point null and point alternative"
  2. "Failing to obtain $\chi^2(1)$ asymptotic distribution under $H_0$ in a likelihood ratio test"
  • $\begingroup$ Likelihood ratio test: en.wikipedia.org/wiki/Likelihood-ratio_test $\endgroup$
    – stans
    Commented Jan 26, 2021 at 8:36
  • $\begingroup$ @stans, looks like a great starting point! This should give me the test statistic in a straightforward way. But how do I find the rejection region and the $p$-value? Can I simply (i) find the likelihood maximizer among $\{0,2\}$, denote it $\hat\mu_0$, (ii) find the likelihood maximizer among $\{1,3\}$, denote it $\hat\mu_1$ and (iii) proceed as if $H_0\colon \mu=\hat\mu_0$ and $H_1\colon \mu=\hat\mu_1$? Could you recommend a textbook that introduces LR testing in an accessible way and treats cases like mine in addition to the simplest cases? $\endgroup$ Commented Jan 26, 2021 at 8:50
  • $\begingroup$ I guess, you could simulate the distribution of the exact LR statistic under the conservative subcase of the null hypothesis. By "conservative" I mean the one which is harder to detect: $\mu = 2$. Then the rejection region is the right tail of the distribution of LR statistic... Regarding the ML estimates of $\mu$, they want you to find i) the estimate on $\{0, 2\}$ and ii) the estimate on $\{0, 1, 2, 3\}$... I learned LR test from Lehman's "Theory of Point Estimation" but there are probably friendlier books. Unfortunately, I do not know of a reference tackling your exact problem. $\endgroup$
    – stans
    Commented Jan 26, 2021 at 9:04
  • $\begingroup$ Did I understand your question correctly? $\endgroup$
    – stans
    Commented Jan 26, 2021 at 9:04
  • 1
    $\begingroup$ You're welcome, Richard. $\endgroup$
    – stans
    Commented Jan 26, 2021 at 10:15

1 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.

![enter image description here

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 $\sigma=1$.


  • 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.

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