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The terms "type I" (or "alpha) and "type II" (or "beta) error, to denote false positive and false negative, are often used. What is the history of those terms?

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Neyman and Pearson were the pioneers in formulating the integral framework of the modern theory of hypothesis testing that we are acquainted with. While I cannot assert with absolute certainty, it is plausible their works were instrumental in developing or, for the matter, popularising the nomenclatures in question.

From [I]:

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From [II]:

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From [III]:

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References:

[I] J. NEYMAN, PH.D, E. S. PEARSON, D.Sc, ON THE USE AND INTERPRETATION OF CERTAIN TEST CRITERIA FOR PURPOSES OF STATISTICAL INFERENCE PART I, Biometrika, Volume 20A, Issue 1-2, December 1928, Pages 175–240, https://doi.org/10.1093/biomet/20A.1-2.175.

[II] IX. On the problem of the most efficient tests of statistical hypotheses, Jerzy Neyman, Egon Sharpe Pearson, 16 February 1933, https://doi.org/10.1098/rsta.1933.0009.

[III] Neyman, J., & Pearson, E. (1933). The testing of statistical hypotheses in relation to probabilities a priori. Mathematical Proceedings of the Cambridge Philosophical Society, 29(4), 492-510. doi:10.1017/S030500410001152X.

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    $\begingroup$ Yep, Miller's list -- mathshistory.st-andrews.ac.uk/Miller/mathword/t agrees, citing David (1995) (which would be David, H. A. (1995), "First (?) Occurrence of Common Terms in Mathematical Statistics," The American Statistician, 49, 121-133; there's also a 1998 update by David in the same journal) $\endgroup$
    – Glen_b
    Commented Aug 22, 2022 at 5:12
  • $\begingroup$ Very well, thanks, for pointing that out @Glen_b. $\endgroup$ Commented Aug 22, 2022 at 5:15
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    $\begingroup$ Calling these "errors" has resulted in a hundred years of unclear thinking. They are not error probabilities but merely assertion trigger probabilities. For a full explanation see hbiostat.org/bbr/alpha.html. What's needed is the probability of a decision error which has nothing to do with $\alpha$ and $\beta$. $\endgroup$ Commented Aug 22, 2022 at 11:48
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    $\begingroup$ I agree. That is unfortunate @FrankHarrell. I have come across rather mild discontentment from various authors: traditionally "non-mnemonic" as written by Casella, Berger; Lehmann seldom used the two terms in his treatise. But the tradition is deeply rooted. $\endgroup$ Commented Aug 22, 2022 at 12:24

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