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.

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.