Log-t distribution

In probability theory, a log-t distribution or log-Student t distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Student's t-distribution. If X is a random variable with a Student's t-distribution, then Y = exp(X) has a log-t distribution; likewise, if Y has a log-1distribution, then X = log(Y) has a Student's t-distribution.

The log-t distribution has the probability density function:

$p(x\mid \nu ,{\hat {\mu }},{\hat {\sigma }})={\frac {\Gamma ({\frac {\nu +1}{2}})}{x\Gamma ({\frac {\nu }{2}}){\sqrt {\pi \nu }}{\hat {\sigma }}\,}}\left(1+{\frac {1}{\nu }}\left({\frac {\ln x-{\hat {\mu }}}{\hat {\sigma }}}\right)^{2}\right)^{-{\frac {\nu +1}{2}}}$,

where \$hat{\mu }$ is the location parameter of the underlying (non-standardized) Student's t-distribution, ${\hat {\sigma }}$ is the scale parameter of the underlying (non-standardized) Student's t-distribution, and $\nu$ is the number of degrees of freedom of the underlying Student's t-distribution.