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Julian Faraway, in his book on linear model with R, used half normal plot to find hat(a value in hat matrix) with too large value.

That's the only example where I saw half normal distribution is used. But other than that, I can't come up with real application of it though I guess it might be useful for approximating some positive numbers whose mean is near zero.

What are real applications of it?

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    $\begingroup$ Because the square root of a $\chi^2(1)$ variable is half normal, most applications involving chi squared values with 1 degree of freedom can be interpreted as applications of a half-normal model. $\endgroup$
    – whuber
    Dec 27, 2011 at 21:10
  • $\begingroup$ The half normal is used in the EGARCH specification for asymmetric volatility modelling. $\endgroup$
    – user11978
    Jun 14, 2012 at 3:41

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In Bayesian statistics the half normal,with a sufficiently large variation parameter, can be used as a noninformative prior distribution on the SD of a standard distribution. This is suggested in, for example:

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian analysis, 1, 515-534.

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    $\begingroup$ It's true that you use it, but if you do it's typically NOT noninformative, since then you could also use a more typical/easier noninformative prior like gamma or the uniform. The half-t makes more sense as a weakly informative prior, which has a distinct tendency but wide enough tails to let itself be overwhelmed by the data in the face of strong evidence. $\endgroup$
    – Erik
    Nov 13, 2012 at 12:00
  • $\begingroup$ i have came across a function that uses a single value as a paramter for the half normal distribution in a Baysian Market Media Mix context - stats.stackexchange.com/questions/599740/… - and i simply cannot figure out how that is supposed to work. Maybe you can explain? $\endgroup$
    – Nneka
    Dec 22, 2022 at 9:17
  • $\begingroup$ @Nneka Often that's actually a half-normal centered at zero. Then the half-normal can be defined by either it's SD (would it have been a full normal) or the mean or median of the half-normal. $\endgroup$ Dec 23, 2022 at 15:19
  • $\begingroup$ @RasmusBååth does that mean that i can actually define the half-normal centered at zero if i only have the mean? How could i do that ? $\endgroup$
    – Nneka
    Jan 2, 2023 at 10:44
  • $\begingroup$ also here quora.com/… it says that the mean of a half normal distribution centred at zero is the min(X)? $\endgroup$
    – Nneka
    Jan 2, 2023 at 13:30
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In quality control, there is something called a moving-range statistic which is the absolute value of successive differences. The half-normal serves as the basis of the chart as discussed in the following article:

http://www.tandfonline.com/doi/abs/10.1080/08982119508904612#preview

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