Log-t distribution

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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]

Quick Facts Parameters, Support ...
Log-t or Log-Student t
Parameters (real), location parameter
(real), scale parameter
(real), degrees of freedom (shape) parameter
Support
PDF
Mean infinite
Median
Variance infinite
Skewness does not exist
Excess kurtosis does not exist
MGF does not exist
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Characterization

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Perspective

The log-t distribution has the probability density function:

,

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.

If then the distribution is a log-Cauchy distribution.[1] As 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

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