What is the PDF for a log-log-normal distribution?

A log-log-normal distribution is a continuous probability distribution of a random variable whose logarithm logarithm $$\ln(\ln(x))$$ is normally distributed.

What is the Probability Density Function for a log-log-normal distribution?

I could find an equation on page 27 (expression 2.3) of this PhD thesis but I am not sure about the $$\kappa$$ parameter related to attenuation. Is it always there, and what is it called? Also, the variable in the example is between 0 and 1, but I wonder what the function would be for variables greater than 1.

I also found this dissertation, but it is not available online. I wonder if there are other online materials that could be useful to study this distribution.

• An exhaustive search turns up nothing in your first reference that mentions a "log-log-normal" distribution, so please tell us explicitly what you are referring to. – whuber Jan 14 at 21:29
• Expression 2.3 on page 27. – Ismael Ghalimi Jan 14 at 21:31
• That explains why I couldn't find it--it is called an "LLN" distribution! But since equation (2.3) gives the PDF, why are you asking what it is? The function isn't defined for $g \ge 1,$ as you can check (from the fact that $\kappa$ is negative). Evidently $\kappa$ is a unit conversion factor to decibels. – whuber Jan 14 at 21:34
• $\kappa$ is not a "parameter" in the statistical sense: it merely is a unit conversion used for engineering. It's superfluous as far as statistical or mathematical analysis might be concerned. (After all, it can be absorbed into the definition of $m_c$.) There is no "alternative definition" because of the limited domain of the logarithm. – whuber Jan 14 at 21:40
• I don't think so: all the minus signs I see are needed. You could define a distribution on $g\gt 1$ because $\log\log(g)$ would then be defined, but this wouldn't be an essentially different one because $\log\log(g) = \log(-\log(1/g)),$ allowing you always to assume $g\lt 1.$ – whuber Jan 14 at 21:44

With $$\mu$$ and $$\sigma$$ being the mean and standard deviation of the underlying normal process:
$$f(x) = \displaystyle \frac{1}{\sqrt{2\pi\sigma}x\ln(x)}\exp\Bigg({\frac{-\big(\ln(\ln(x)) - \mu\big)^2}{2\sigma^2}}\Bigg) \quad x \geq 1$$
• You need a minus sign in front of the $\log x$ term. – whuber Jan 14 at 22:22