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.
Best Answer
With $\mu$ and $\sigma$ being the mean and standard deviation of the underlying normal process:
$f(x) = \displaystyle \frac{1}{\sqrt{2\pi\sigma}x\ln(x)}\exp\Bigg({\frac{-\big(\ln(\ln(x)) - \mu\big)^2}{2\sigma^2}}\Bigg) \quad x \geq 1$