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 $\nu$ degrees of freedom; its probability density function is given by:Template:Citation needed

$g\left(x\right)=\frac{\mathrm{\Gamma }\left(\frac{\nu +1}{2}\right)}{\mathrm{\Gamma }\left(\frac{\nu }{2}\right)\sqrt{\nu \pi {\sigma }^2}}\{{[1+\frac{1}{\nu }\frac{{(x-\mu )}^2}{{\sigma }^2}]}^{-\frac{\nu +1}{2}}+{[1+\frac{1}{\nu }\frac{{(x+\mu )}^2}{{\sigma }^2}]}^{-\frac{\nu +1}{2}}\}(\text{for}x\ge 0)$.

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

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

$g\left(x\right)=\frac{2\mathrm{\Gamma }\left(\frac{\nu +1}{2}\right)}{\mathrm{\Gamma }\left(\frac{\nu }{2}\right)\sqrt{\nu \pi {\sigma }^2}}{(1+\frac{1}{\nu }\frac{x^2}{{\sigma }^2})}^{-\frac{\nu +1}{2}}(\text{for}x\ge 0)$.

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

$\mathrm{E}[X]=2\sigma \sqrt{\frac{\nu }{\pi }}\frac{\mathrm{\Gamma }(\frac{\nu +1}{2})}{\mathrm{\Gamma }(\frac{\nu }{2})(\nu -1)}\text{for}\nu >1$,

and

$\mathrm{Var}(X)={\sigma }^2(\frac{\nu }{\nu -2}-\frac{4\nu }{\pi (\nu -1{)}^2}{\left(\frac{\mathrm{\Gamma }(\frac{\nu +1}{2})}{\mathrm{\Gamma }(\frac{\nu }{2})}\right)}^2)\text{for}\nu >2$.

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 $\nu =1$.

See also

• Folded normal distribution
• Half-normal distribution
• Modified half-normal distribution
• Half-logistic distribution

Further reading

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External links

• Functions to evaluate half-t distributions are available in several R packages, e.g. [1] [2] [3].

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