Log-t distribution

Log-t or Log-Student t

Parameters	[math]\displaystyle{ \hat{\mu} }[/math] (real), location parameter [math]\displaystyle{ \displaystyle \hat{\sigma} \gt 0\! }[/math] (real), scale parameter [math]\displaystyle{ \nu }[/math] (real), degrees of freedom (shape) parameter
Support	[math]\displaystyle{ \displaystyle x \in (0, +\infty)\! }[/math]
PDF	[math]\displaystyle{ 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}} }[/math]
Mean	infinite
Median	[math]\displaystyle{ e^\hat{\mu}\, }[/math]
Variance	infinite
Skewness	does not exist
Kurtosis	does not exist
MGF	does not exist

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:

[math]\displaystyle{ 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}} }[/math],

where [math]\displaystyle{ \hat{\mu} }[/math] is the location parameter of the underlying (non-standardized) Student's t-distribution, [math]\displaystyle{ \hat{\sigma} }[/math] is the scale parameter of the underlying (non-standardized) Student's t-distribution, and [math]\displaystyle{ \nu }[/math] is the number of degrees of freedom of the underlying Student's t-distribution. If [math]\displaystyle{ \hat{\mu}=0 }[/math] and [math]\displaystyle{ \hat{\sigma}=1 }[/math] then the underlying distribution is the standardized Student's t-distribution.

If [math]\displaystyle{ \nu=1 }[/math] then the distribution is a log-Cauchy distribution. As [math]\displaystyle{ \nu }[/math] 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.