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-t distribution, then X = log(Y) has a Student's t-distribution.

Log-t or Log-Student t

Parameters	${\hat {\mu }}$ (real), location parameter $\displaystyle {\hat {\sigma }}>0\!$ (real), scale parameter $\nu$ (real), degrees of freedom (shape) parameter
Support	$\displaystyle x\in (0,+\infty )\!$
PDF	$p(x\mid \nu ,{\hat {\mu }},{\hat {\sigma }})={\frac {\Gamma ({\frac {\nu +1}{2}})}{x\Gamma ({\frac {\nu }{2}}){\sqrt {\pi \nu }}{\hat {\sigma }}\,}}\left(1+{\frac {1}{\nu }}\left({\frac {\ln x-{\hat {\mu }}}{\hat {\sigma }}}\right)^{2}\right)^{-{\frac {\nu +1}{2}}}$
Mean	infinite
Median	$e^{\hat {\mu }}\,$
Variance	infinite
Skewness	does not exist
Excess kurtosis	does not exist
MGF	does not exist

Characterization

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. If ${\hat {\mu }}=0$ and ${\hat {\sigma }}=1$ then the underlying distribution is the standardized Student's t-distribution.

If $\nu =1$ then the distribution is a log-Cauchy distribution. As $\nu$ approaches infinity, the distribution approaches a log-normal distribution. Although the log-normal distribution has finite moments, for any finite degrees of freedom, the mean and variance and all higher moments of the log-t distribution are infinite or do not exist.

The log-t distribution is a special case of the generalized beta distribution of the second kind. The log-t distribution is an example of a compound probability distribution between the lognormal distribution and inverse gamma distribution whereby the variance parameter of the lognormal distribution is a random variable distributed according to an inverse gamma distribution.

Applications

The log-t distribution has applications in finance. For example, the distribution of stock market returns often shows fatter tails than a normal distribution, and thus tends to fit a Student's t-distribution better than a normal distribution. While the Black-Scholes model based on the log-normal distribution is often used to price stock options, option pricing formulas based on the log-t distribution can be a preferable alternative if the returns have fat tails. The fact that the log-t distribution has infinite mean is a problem when using it to value options, but there are techniques to overcome that limitation, such as by truncating the probability density function at some arbitrary large value.

The log-t distribution also has applications in hydrology and in analyzing data on cancer remission.

Multivariate log-t distribution

Analogous to the log-normal distribution, multivariate forms of the log-t distribution exist. In this case, the location parameter is replaced by a vector μ, the scale parameter is replaced by a matrix Σ.