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]
where is the
location parameter of the underlying (non-standardized) Student's t-distribution, is the
scale parameter of the underlying (non-standardized) Student's t-distribution, and is the number of
degrees of freedom of the underlying Student's t-distribution.[1] If and then the underlying distribution is the standardized Student's t-distribution.
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]
^Marshall, Albert W.; Olkin, Ingram (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer. p. 445.
ISBN978-1921209680.