Log-t distribution
Parameters |
(real), location parameter (real), scale parameter (real), degrees of freedom (shape) parameter | ||
---|---|---|---|
Support | |||
Mean | infinite | ||
Median | |||
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:
- ,
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
Applications
The log-t distribution has applications in finance.
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,
References
- ^ doi:10.15446/rce.v45n1.90672. Retrieved 2022-04-01.)
{{cite journal}}
: CS1 maint: multiple names: authors list (link - ISBN 978-1921209680.
- ^ JSTOR 2352878. Retrieved 2022-04-05.
- JSTOR 1927230.
- .
- ^ S2CID 100313689.
- JSTOR 822677. Retrieved 2022-04-05.
- S2CID 121129552.
- . Retrieved 2022-04-01.