 ;/math ;
 | cdf =
 | qf =
 | qdf =
 | mean =infinite 
 | median =formula_1
 | mode =
 | variance =infinite
 | skewness =does not exist
 | kurtosis =does not exist
 | entropy =
 | mgf =does not exist
 | char =
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:
formula_2,
where formula_3 is the location parameter of the underlying (non-standardized) Student's t-distribution, formula_4 is the scale parameter of the underlying (non-standardized) Student's t-distribution, and formula_5 is the number of degrees of freedom of the underlying Student's t-distribution. If formula_6 and formula_7 then the underlying distribution is the standardized Student's t-distribution.
If formula_8 then the distribution is a log-Cauchy distribution. As formula_5 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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