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If the log of a dataset fits a normal distribution, then the data is said to be log normal. If the log of a dataset fits a log normal distribution, is the data said to be log log normal?

Is there a more appropriate name than a 'log log normal' distribution?

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    $\begingroup$ For the logarithm of a random variable $X$ to follow a log-normal distribution, it must be the case the random variable has its support constraint to the $X\in (1,\infty)$ interval. is this your case? $\endgroup$ Commented Nov 28, 2015 at 21:35
  • $\begingroup$ Regarding your title: it's not the distribution that's being exponentiated, in fact the object being exponentiated is the random variable. You actually want a name for the distribution of an exponentiated lognormal-variate. $\endgroup$
    – Glen_b
    Commented Nov 29, 2015 at 7:37

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There's no "standard name" for this distribution, but the intent of "log-log-normal" would probably be correctly guessed at.

Consequently if you must have a name for it*, calling it that (hopefully with an explanation at the time of first use) would probably be sufficient.

* people seem to love to name things, as if the unnamed thing could not be clearly understood but once named it is treated as if it were a known quantity. I don't know why we tend to think like that but it's clear that humans often do; I catch myself at it sometimes.

Once they are counted and compelled,
They can quickly be dispelled.

-- Jerry Juhl, Fears of Zero

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  • $\begingroup$ Naming provides power over the named. Brothers Grim tale Rumpelstiltskin, en.wikipedia.org/wiki/Rumpelstiltskin, is a more modern version of this ancient belief (or fact). $\endgroup$ Commented Nov 29, 2015 at 0:13
  • $\begingroup$ @Alecos thanks, yes, I am aware of the historical/cultural belief (and can point to numerous instances of it in that sense, from true names of demons giving power over them to various other ideas, and the quote was meant as an indirect reference to it) but it runs deeper than that - even in modern times among people who appear not in the least superstitious, seeming to speak to something in the way our minds work. $\endgroup$
    – Glen_b
    Commented Nov 29, 2015 at 0:15
  • $\begingroup$ Incidentally are there any natural phenomenon that could be modelled by a log-log-normal distribution ? $\endgroup$ Commented Nov 29, 2015 at 13:45
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Per this article, if you call the normal distribution “additive normal” and the log-normal distribution “multiplicative normal”, the distribution you wrote about would be called “expansive normal”. And here is its PDF.

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