Log-t distribution explained

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:

p(x\mid \nu,\hat{\mu},\hat{\sigma}) = (\Gamma( (\nu +1)/() 2))/(x\Gamma( (\nu)/()\sqrt{\pi\nu 2)) \hat\sigma} \left( 1+ (1)/(\nu) \left( (ln x - \hat{\mu)/(}) {\hat{\sigma} } \right) ^{2\right)} ^{- (\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.^[1] 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.^[1] As \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]

Notes and References

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