In null hypothesis statistical testing, the critical region (also known as the rejection region) is

A set of values for the test statistic for which the null hypothesis is rejected. i.e. if the observed test statistic is in the critical region then we reject the null hypothesis and accept the alternative hypothesis. [1]

For a Stats 101-like course, they might be illustrated as follow (omitting most of the details):

Critical values and critical/rejection region for a test statistic

The question is: How do you call the complement of the critical region / rejection region (the interval where the distribution is not shaded)? Is there a specific term for this interval?

I have heard of the interval being referred to as "non-rejection region" and "acceptance region" (which IMO is misleading as it implies acceptance of the null), though I am looking for answers with reference(s) that have defined / discussed on such terminology.

Note: I have seen this question but the question deals with the Bayesian version.

[1] Newcastle University (2018) Critical Region and Confidence Interval. https://internal.ncl.ac.uk/ask/numeracy-maths-statistics/statistics/hypothesis-testing/critical-region-and-confidence-interval.html

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    $\begingroup$ Are you inquiring what a set complement means? $\endgroup$ – whuber Mar 12 at 19:03
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    $\begingroup$ @whuber - Nope, I want to know a concise name of the complement for the critical region / rejection region (instead of referring it as e.g. "the set complement of the interval referred as critical region") - will take suggestion on how to improve the question! $\endgroup$ – B.Liu Mar 12 at 19:06
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    $\begingroup$ Thank you clarifying and emphasizing this is a terminological question. (Some of us read questions too hastily ;-).) $\endgroup$ – whuber Mar 12 at 19:09

Acceptance region. There's plenty more references in the web.

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    $\begingroup$ That's an awful name, since failing to reject $\text{H}_{0}$ may result either because the null is true, or because $\text{H}_{\text{A}}$ is true, but there is not enough power to reject. Also, this is why combining tests for difference with tests for equivalence is dope. :) $\endgroup$ – Alexis Mar 12 at 19:29
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    $\begingroup$ While this is what Casella calls it, we all agreed that this terminology is poor, as we do not accept null hypotheses. $\endgroup$ – Dave Mar 12 at 19:29
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    $\begingroup$ Since we don't accept the null, but we need a word that is the opposite of critical, I propose we call it the 'insincere flattery region', @Alexis ;-P. $\endgroup$ – gung - Reinstate Monica Mar 12 at 19:35
  • $\begingroup$ @Dave, we do not, that's true, but we say that a lot, because it's shorter. I guess names and labels are like models: never really right, but still useful $\endgroup$ – carlo Mar 12 at 20:11
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    $\begingroup$ I am fine with calling it the acceptance region when statisticians are talking, since probably all of us read Casella and had the professor say, "Isn't that name awful? But it is what it is." However, when you have to explain in a meeting why we don't accept the null hypothesis when the value falls in the acceptance region... $\endgroup$ – Dave Mar 12 at 20:15

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