2
$\begingroup$

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"
$\endgroup$
14
  • $\begingroup$ Likelihood ratio test: en.wikipedia.org/wiki/Likelihood-ratio_test $\endgroup$
    – stans
    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? $\endgroup$ 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
    Jan 26, 2021 at 9:04
  • $\begingroup$ Did I understand your question correctly? $\endgroup$
    – stans
    Jan 26, 2021 at 9:04
  • 1
    $\begingroup$ You're welcome, Richard. $\endgroup$
    – stans
    Jan 26, 2021 at 10:15

1 Answer 1

2
$\begingroup$

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 $\sigma=1$.

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.