Folded-t and half-t distributions explained
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
degrees of freedom; its
probability density function is given by:
| g\left(x\right) = | | \Gamma\left( | \nu+1 | \right) | | 2 |
|
| \Gamma\left( | \nu | \right)\sqrt{\nu\pi\sigma2 | | 2 |
|
}\left\lbrace\left[1+\frac{1}{\nu}\frac{\left(x-\mu\right)^2}{\sigma^2}\right]^+\left[1+\frac{1}{\nu}\frac{\left(x+\mu\right)^2}{\sigma^2}\right]^ \right\rbrace \qquad(\mbox\quad x \geq 0).The
half-t distribution results as the special case of
, and the standardized version as the special case of
.
If
, the folded-
t distribution reduces to the special case of the half-
t distribution. Its
probability density function then simplifies to
| g\left(x\right) = | | 2 \Gamma\left( | \nu+1 | \right) | | 2 |
|
| \Gamma\left( | \nu | \right)\sqrt{\nu\pi\sigma2 | | 2 |
|
}\left(1+\frac\frac\right)^ \qquad(\mbox\quad x \geq 0).The half-
t distribution's first two
moments (
expectation and
variance) are given by:
| \operatorname{E}[X] = 2\sigma\sqrt{ | \nu |
| \pi |
}\frac \qquad\mbox\quad \nu > 1,and
| | |
| \operatorname{Var}(X) = \sigma | | - |
\right)2\right) for \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
.
See also
Further reading
- Psarakis. S.. Panaretos. J.. The folded t distribution. Communications in Statistics - Theory and Methods. 19. 7. 1990. 2717–2734. 10.1080/03610929008830342. 121332770 .
- Gelman. A.. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis. 1. 3. 515–534. 2006. 10.1214/06-BA117A . free.
- M. P. . Wiper . F. J. . Girón . Arthur . Pewsey . Objective Bayesian Inference for the Half-Normal and Half-t Distributions . Communications in Statistics - Theory and Methods . 37 . 2008 . 20 . 3165–3185 . 10.1080/03610920802105184 . 117937250 .
- Tancredi . A. . 2002 . Accounting for heavy tails in stochastic frontier models . Università degli Studi di Padova . 7325 . Working paper.
External links
