• Source: 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
∣
ν
,



μ
^



,



σ
^



)
=



Γ
(



ν
+
1

2


)


x
Γ
(


ν
2


)


π
ν





σ
^









(

1
+


1
ν




(



ln
⁡
x
−



μ
^






σ
^




)


2



)


−



ν
+
1

2






{\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}}}}

,
where






μ
^





{\displaystyle {\hat {\mu }}}

is the location parameter of the underlying (non-standardized) Student's t-distribution,






σ
^





{\displaystyle {\hat {\sigma }}}

is the scale parameter of the underlying (non-standardized) Student's t-distribution, and



ν


{\displaystyle \nu }

is the number of degrees of freedom of the underlying Student's t-distribution. If






μ
^



=
0


{\displaystyle {\hat {\mu }}=0}

and






σ
^



=
1


{\displaystyle {\hat {\sigma }}=1}

then the underlying distribution is the standardized Student's t-distribution.
If



ν
=
1


{\displaystyle \nu =1}

then the distribution is a log-Cauchy distribution. As



ν


{\displaystyle \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.




References

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