Log-t distribution

Log-t or Log-Student t

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

Characterization

The log-t distribution has the probability density function:

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

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

If ${\displaystyle \nu =1}$ then the distribution is a log-Cauchy distribution.^[1] As ${\displaystyle \nu }$ approaches infinity, the distribution approaches a log-normal distribution.^[1][2] 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.^[1]

The log-t distribution is a special case of the generalized beta distribution of the second kind.^[1][3][4] 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.^[3][5]

Applications

The log-t distribution has applications in finance.^[3] 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.^[6] 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.^[6][7][8]

The log-t distribution also has applications in hydrology and in analyzing data on cancer remission.^[1][9]

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.^[1]

References

1. Olosunde, Akinlolu & Olofintuade, Sylvester (January 2022). "Some Inferential Problems from Log Student's T-distribution and its Multivariate Extension". Revista Colombiana de Estadística - Applied Statistics. 45 (1): 209–229. doi:10.15446/rce.v45n1.90672. Retrieved 2022-04-01.
2. Marshall, Albert W.; Olkin, Ingram (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer. p. 445. ISBN 978-1921209680.
3. Bookstaber, Richard M.; McDonald, James B. (July 1987). "A General Distribution for Describing Security Price Returns". The Journal of Business. 60 (3). University of Chicago Press: 401–424. doi:10.1086/296404. JSTOR 2352878. Retrieved 2022-04-05.
4. McDonald, James B.; Butler, Richard J. (May 1987). "Some Generalized Mixture Distributions with an Application to Unemployment Duration". The Review of Economics and Statistics. 69 (2): 232–240. doi:10.2307/1927230. JSTOR 1927230.
5. Vanegas, Luis Hernando; Paula, Gilberto A. (2016). "Log-symmetric distributions: Statistical properties and parameter estimation". Brazilian Journal of Probability and Statistics. 30 (2): 196–220. doi:10.1214/14-BJPS272.
6. Cassidy, Daniel T.; Hamp, Michael J.; Ouyed, Rachid (2010). "Pricing European Options with a Log Student's t-Distribution: a Gosset Formula". Physica A. 389 (24): 5736–5748. arXiv:0906.4092. Bibcode:2010PhyA..389.5736C. doi:10.1016/j.physa.2010.08.037. S2CID 100313689.
7. Kou, S.G. (August 2022). "A Jump-Diffusion Model for Option Pricing". Management Science. 48 (8): 1086–1101. doi:10.1287/mnsc.48.8.1086.166. JSTOR 822677. Retrieved 2022-04-05.
8. Basnarkov, Lasko; Stojkoski, Viktor; Utkovski, Zoran; Kocarev, Ljupco (2019). "Option Pricing with Heavy-tailed Distributions of Logarithmic Returns". International Journal of Theoretical and Applied Finance. 22 (7). arXiv:1807.01756. doi:10.1142/S0219024919500419. S2CID 121129552.
9. Viglione, A. (2010). "On the sampling distribution of the coefficient of L-variation for hydrological applications" (PDF). Hydrology and Earth System Sciences Discussions. 7: 5467–5496. doi:10.5194/hessd-7-5467-2010. Retrieved 2022-04-01.