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 ν {\displaystyle \nu } degrees of freedom; its probability density function is given by:
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 {\displaystyle \mu =0} , and the standardized version as the special case of σ = 1 {\displaystyle \sigma =1} . If μ = 0 {\displaystyle \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:
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 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 {\displaystyle \nu =1} .
See also Folded normal distribution Half-normal distribution Modified half-normal distribution Half-logistic distribution
References
Further reading 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 Functions to evaluate half-t distributions are available in several R packages, e.g. [1] [2] [3].