Folded-t and half-t distributions

The folded non-standardized t distribution is the distribution of the absolute value of the non-standardized t distribution with $\backslash nu$ degrees of freedom; its probability density function is given by:{{citation needed|date=November 2016}}

:$g\backslash left(x\backslash right)\backslash ;=\backslash ;\backslash frac\{\backslash Gamma\backslash left(\backslash frac\{\backslash nu+1\}\{2\}\backslash right)\}\{\backslash Gamma\backslash left(\backslash frac\{\backslash nu\}\{2\}\backslash right)\backslash sqrt\{\backslash nu\backslash pi\backslash sigma^2\}\}\backslash left\backslash lbrace$

\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 $\backslash mu=0$, and the standardized version as the special case of $\backslash sigma=1$.

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

:$g\backslash left(x\backslash right)\backslash ;=\backslash ;\backslash frac\{2\backslash ;\backslash Gamma\backslash left(\backslash frac\{\backslash nu+1\}\{2\}\backslash right)\}\{\backslash Gamma\backslash left(\backslash frac\{\backslash nu\}\{2\}\backslash right)\backslash sqrt\{\backslash nu\backslash pi\backslash sigma^2\}\}$

\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:{{citation|last1=Psarakis|first1=S.|last2=Panaretos|first2=J.|title=The folded t distribution|journal=Communications in Statistics - Theory and Methods|volume=19|issue=7|year=1990|pages=2717–2734|doi=10.1080/03610929008830342|s2cid=121332770 }}

:$\backslash operatorname\{E\}[X]\backslash ;=\backslash ;2\backslash sigma\backslash sqrt\{\backslash frac\{\backslash nu\}\{\backslash pi\}\}\backslash frac\{\backslash Gamma(\backslash frac\{\backslash nu+1\}\{2\})\}\{\backslash Gamma(\backslash frac\{\backslash nu\}\{2\})\backslash ,(\backslash nu-1)\}\; \backslash qquad\backslash mbox\{for\}\backslash quad\; \backslash nu\; >\; 1$,

and

:$\backslash operatorname\{Var\}(X)\backslash ;=\backslash ;\backslash sigma^2\backslash left(\backslash frac\{\backslash nu\}\{\backslash nu-2\}-\backslash frac\{4\backslash nu\}\{\backslash pi(\backslash nu-1)^2\}\backslash left(\backslash frac\{\backslash Gamma(\backslash frac\{\backslash nu+1\}\{2\})\}\{\backslash Gamma(\backslash frac\{\backslash nu\}\{2\})\}\backslash right)^2\backslash right)\; \backslash qquad\backslash mbox\{for\}\backslash quad\; \backslash nu\; >\; 2$.

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 $\backslash nu=1$.

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

{{Reflist}}

• {{cite journal |last1=Psarakis|first1=S.|last2=Panaretos|first2=J.|title=The folded t distribution|journal=Communications in Statistics - Theory and Methods|volume=19|issue=7|year=1990|pages=2717–2734|doi=10.1080/03610929008830342|s2cid=121332770 }}
• {{cite journal |last1=Gelman|first1=A.|title=Prior distributions for variance parameters in hierarchical models|journal=Bayesian Analysis|volume=1|number=3|pages=515–534|year=2006|doi=10.1214/06-BA117A |url=http://projecteuclid.org/euclid.ba/1340371048|doi-access=free|url-access=subscription}}
• {{Citation | last1=Röver | first1=C. | last2=Bender | first2=R. | last3=Dias | first3=S. | last4=Schmid | first4=C.H. | last5=Schmidli | first5=H. | last6=Sturtz | first6=S. | last7=Weber | first7=S. | last8=Friede | first8=T. | title=On weakly informative prior distributions for the heterogeneity parameter in Bayesian random‐effects meta‐analysis | journal=Research Synthesis Methods | year=2021 | volume=12 | issue=4 | pages=448–474 | doi=10.1002/jrsm.1475 | pmid=33486828 | arxiv=2007.08352| s2cid=220546288 }}
• {{cite journal |first1=M. P. |last1=Wiper |first2=F. J. |last2=Girón |first3=Arthur |last3=Pewsey |title=Objective Bayesian Inference for the Half-Normal and Half-t Distributions |journal=Communications in Statistics - Theory and Methods |volume=37 |year=2008 |issue=20 |pages=3165–3185 |doi=10.1080/03610920802105184 |s2cid=117937250 }}
• {{cite journal |last1=Tancredi |first1=A. |year=2002 |title=Accounting for heavy tails in stochastic frontier models |url=http://paduaresearch.cab.unipd.it/7325/ |institution=Università degli Studi di Padova |number=7325 |series=Working paper}}

• Functions to evaluate half-t distributions are available in several R packages, e.g. [https://rdrr.io/cran/LaplacesDemon/man/dist.Halft.html] [https://rdrr.io/cran/bayesmeta/man/dhalfnormal.html] [https://rdrr.io/cran/extraDistr/man/HalfT.html].

{{ProbDistributions|continuous-semi-infinite}}

Category:Continuous distributions

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