Folded-t and half-t distributions

In statistics, the folded-t and half-t distributions are derived from Student's t-distribution by taking the absolute values of variates. This is analogous to the folded-normal and the half-normal statistical distributions being derived from the normal distribution.

Definitions

The folded non-standardized t distribution is the distribution of the absolute value of the non-standardized t distribution with degrees of freedom; its probability density function is given by:
The half-t distribution results as the special case of, and the standardized version as the special case of.
If, the folded-t distribution reduces to the special case of the half-t distribution. Its probability density function then simplifies to
The half-t distribution's first two moments are given by:
and

Relation to other distributions

Folded-t and half-t generalize the folded normal and half-normal distributions by allowing for finite degrees of [freedom (statistics)|degrees-of-freedom]. Since the Cauchy distribution constitutes the special case of a Student-t distribution with one degree of freedom, the families of folded and half-t distributions include the folded Cauchy distribution and half-Cauchy distributions for.