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I am reading possibly conflicting statements in different sources.

Sometimes I think I understand that the word "Wald test" refers to

  • any kind of test of whether the value of a parameter (whether in a linear model or not) is significantly removed from a null hypothesis. i.e. the Wald test outputs one of $\{\text {Reject},\text{Not-Reject} \}$ given a rejection threshold $\alpha$ (usually 0.05 or 0.01), of the likelihood that a test statistic $S$ farther away from $S_0$ would be obtained if the null hypothesis is true,

  • The above, but specifically to coefficent parameters in a linear regression model.

  • The above, but specifically taking the population variance $\sigma^2$ as given. (i.e. a $t$ test is not a Wald test).

Which of these is it, or is it something else?

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The last one seems pretty close.

It may help to look at the relationship between the Wald, likelihood ratio, and score test, e.g. https://stats.idre.ucla.edu/other/mult-pkg/faq/general/faqhow-are-the-likelihood-ratio-wald-and-lagrange-multiplier-score-tests-different-andor-similar/

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