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[edit]

The folded non-standardized t distribution is the distribution of the absolute value of the non-standardized t distribution with  ν \nu degrees of freedom; its probability density function is given by:[citation needed]

g ( x ) = Γ ( ν + 1 2 ) Γ ( ν 2 ) ν π σ 2 { [ 1 + 1 ν ( x − μ ) 2 σ 2 ] − ν + 1 2 + [ 1 + 1 ν ( x + μ ) 2 σ 2 ] − ν + 1 2 } ( for x ≥ 0 ) {\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  μ = 0 \mu =0 , and the standardized version as the special case of  σ = 1 {\displaystyle \sigma =1} .

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

g ( x ) = 2 Γ ( ν + 1 2 ) Γ ( ν 2 ) ν π σ 2 ( 1 + 1 ν x 2 σ 2 ) − ν + 1 2 ( for x ≥ 0 ) {\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]

E ⁡ [ X ] = 2 σ ν π Γ ( ν + 1 2 ) Γ ( ν 2 ) ( ν − 1 ) for ν > 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

Var ⁡ ( X ) = σ 2 ( ν ν − 2 − 4 ν π ( ν − 1 ) 2 ( Γ ( ν + 1 2 ) Γ ( ν 2 ) ) 2 ) for ν > 2 {\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[edit]

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  ν = 1 \nu =1 .

## See also[edit]

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

## References[edit]

1. ^ Psarakis, S.; Panaretos, J. (1990), "The folded t distribution", Communications in Statistics - Theory and Methods, 19 (7): 2717–2734, doi:10.1080/03610929008830342, S2CID 121332770

## Further reading[edit]

* Psarakis, S.; Panaretos, J. (1990). "The folded t distribution". Communications in Statistics - Theory and Methods. 19 (7): 2717–2734. doi:10.1080/03610929008830342. S2CID 121332770.
* Gelman, A. (2006). "Prior distributions for variance parameters in hierarchical models". Bayesian Analysis. 1 (3): 515–534. doi:10.1214/06-BA117A.
* Röver, C.; Bender, R.; Dias, S.; Schmid, C.H.; Schmidli, H.; Sturtz, S.; Weber, S.; Friede, T. (2021), "On weakly informative prior distributions for the heterogeneity parameter in Bayesian random‐effects meta‐analysis", Research Synthesis Methods, 12 (4): 448–474, arXiv:2007.08352, doi:10.1002/jrsm.1475, PMID 33486828, S2CID 220546288
* Wiper, M. P.; Girón, F. J.; Pewsey, Arthur (2008). "Objective Bayesian Inference for the Half-Normal and Half-t Distributions". Communications in Statistics - Theory and Methods. 37 (20): 3165–3185. doi:10.1080/03610920802105184. S2CID 117937250.
* Tancredi, A. (2002). "Accounting for heavy tails in stochastic frontier models". Working paper (7325). Università degli Studi di Padova. `{{cite journal}}`: Cite journal requires `|journal=` (help)

## External links[edit]

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

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