Intro to question

I am trying to understand sense of unbiasedness in hypothesis testing - here is the definition:

Consider a statistical hypothesis test of size level $0<\alpha<1$ for testing a null hypothesis $H_0:\theta \in \Theta_0 \subset \Theta$ against an alternative hypothesis $H_1:\theta \in \Theta_1=\Theta - \Theta_0$. The test is considered "unbiased" if its power function $\beta$ satisfies the condition:

$$\beta(\theta)\leq\alpha \hspace{0.3cm}\text{if}\hspace{0.3cm}\theta \in \Theta_0,$$ $$\beta(\theta)\geq\alpha \hspace{0.3cm}\text{if}\hspace{0.3cm}\theta \in \Theta_1.$$

Answer to this question: "Unbiased" hypothesis test --- what does it mean actually? provides an example of biased test:

Now, suppose you were to use that rule (reject if $t>1.645$) to test $\theta=0$ against $\theta\neq0$. The probability that the test will reject will decrease the more negative the true $\theta$, as we shall rarely observe large positive t-ratios in that case. In particular, this test is be biased, as $\beta(\theta)<\alpha$ when $\theta\in\Theta_1\cap(-\infty,0)$.

and graphical explanation which is a plot of power function for this particular test: plot1

The question

In the comment someone asked what can we do about the fact that the test is biased. I think, that the solution is to use test statistic $|T|$ instead of $T$. Then power function will be symmetrical around $0$, something like this:

enter image description here

Can anyone confirm if it is true?

  • $\begingroup$ You should specify the new rejection rule but yes, that approach will work. $\endgroup$
    – Glen_b
    Commented Dec 9, 2021 at 5:44
  • $\begingroup$ As far as understanding test bias goes, does this help any? stats.stackexchange.com/q/541570/805 $\endgroup$
    – Glen_b
    Commented Dec 9, 2021 at 5:52
  • $\begingroup$ @Glen_b thanks, your answer to linked question is really useful $\endgroup$ Commented Dec 9, 2021 at 15:11

1 Answer 1


Although this is a good theoretical knowledge, but I have to say you will never design a study to have a $\text{power}<\alpha$ ($=0.05 $ conventionally). Another thing in this explanation, I would like to mention whether $\beta(\theta)$ means type-II error rate or the actual power (although in the text above it says power). Usually $\beta(\theta)$ is used to denote type-II error rate; If $\beta(\theta)$ is the type-II error rate, then $\text{power}=1-\beta(\theta)$.

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    $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$
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    Commented Jan 4 at 13:56

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