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 [math]\displaystyle{ \nu }[/math] degrees of freedom; its probability density function is given by:

[math]\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) }[/math].

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

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

[math]\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) }[/math].

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

[math]\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 \gt 1 }[/math],

and

[math]\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 \gt 2 }[/math].

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 [math]\displaystyle{ \nu=1 }[/math].

See also

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

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. 
• Gelman, A. (2006). "Prior distributions for variance parameters in hierarchical models". Bayesian Analysis 1 (3): 515–534. doi:10.1214/06-BA117A. http://projecteuclid.org/euclid.ba/1340371048. 
• 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, doi:10.1002/jrsm.1475, PMID 33486828 
• 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. 
• Tancredi, A. (2002). Accounting for heavy tails in stochastic frontier models. Working paper. http://paduaresearch.cab.unipd.it/7325/. 

External links

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