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 ${\displaystyle \nu }$ degrees of freedom; its probability density function is given by:^{[citation needed]}

${\displaystyle g\left(x\right)\;=\;{\frac {\Gamma \left({\frac {\nu +1}{2))\right)}{\Gamma \left({\frac {\nu }{2))\right){\sqrt {\nu \pi \sigma ^{2))))}\left\lbrace \left[1+{\frac {1}{\nu )){\frac {\left(x-\mu \right)^{2)){\sigma ^{2))}\right]^{-{\frac {\nu +1}{2))}+\left[1+{\frac {1}{\nu )){\frac {\left(x+\mu \right)^{2)){\sigma ^{2))}\right]^{-{\frac {\nu +1}{2))}\right\rbrace \qquad ({\mbox{for))\quad x\geq 0)}$.

The half-t distribution results as the special case of ${\displaystyle \mu =0}$, and the standardized version as the special case of ${\displaystyle \sigma =1}$.

If ${\displaystyle \mu =0}$, the folded-t distribution reduces to the special case of the half-t distribution. Its probability density function then simplifies to

${\displaystyle g\left(x\right)\;=\;{\frac {2\;\Gamma \left({\frac {\nu +1}{2))\right)}{\Gamma \left({\frac {\nu }{2))\right){\sqrt {\nu \pi \sigma ^{2))))}\left(1+{\frac {1}{\nu )){\frac {x^{2)){\sigma ^{2))}\right)^{-{\frac {\nu +1}{2))}\qquad ({\mbox{for))\quad x\geq 0)}$.

The half-t distribution's first two moments (expectation and variance) are given by:^[1]

${\displaystyle \operatorname {E} [X]\;=\;2\sigma {\sqrt {\frac {\nu }{\pi ))}{\frac {\Gamma ({\frac {\nu +1}{2)))}{\Gamma ({\frac {\nu }{2)))\,(\nu -1)))\qquad {\mbox{for))\quad \nu >1}$,

and

${\displaystyle \operatorname {Var} (X)\;=\;\sigma ^{2}\left({\frac {\nu }{\nu -2))-{\frac {4\nu }{\pi (\nu -1)^{2))}\left({\frac {\Gamma ({\frac {\nu +1}{2)))}{\Gamma ({\frac {\nu }{2)))))\right)^{2}\right)\qquad {\mbox{for))\quad \nu >2}$.

Relation to other distributions

Folded-t and half-t generalize the folded normal and half-normal distributions by allowing for finite degrees-of-freedom (the normal analogues constitute the limiting cases of infinite 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 ${\displaystyle \nu =1}$.