Folded-t and half-t distributions

In statistics, the folded-t and half-t distributions are derived from

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

with one degree of freedom, the families of folded and half-t distributions include the folded Cauchy distribution and half-Cauchy distributions for ${\displaystyle \nu =1}$.

See also

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

References

1. S2CID 121332770

Further reading

• Psarakis, S.; Panaretos, J. (1990). "The folded t distribution". Communications in Statistics - Theory and Methods. 19 (7): 2717–2734.
  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,
  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.
  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

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

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