Half-t Distribution

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 $\nu$ degrees of freedom; its probability density function is given by:
:g\left(x\right)\;=\;\frac\left\lbrace
\left +\frac\frac\right+\left +\frac\frac\right \right\rbrace \qquad(\mbox\quad x \geq 0).
The half-''t'' distribution results as the special case of $\mu=0$, and the standardized version as the special case of $\sigma=1$.

If $\mu=0$, the folded-''t'' distribution reduces to the special case of the half-''t'' distribution. Its probability density function then simplifies to
:$g\left(x\right)\;=\;\frac \left(1+\frac\frac\right)^ \qquad(\mbox\quad x \geq 0)$.
The half-''t'' distribution's first two moments (expectation and variance) are given by:
:\operatorname ;=\;2\sigma\sqrt\frac \qquad\mbox\quad \nu > 1,
and
:$\operatorname(X)\;=\;\sigma^2\left(\frac-\frac\left(\frac\right)^2\right) \qquad\mbox\quad \nu > 2$.

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 $\nu=1$.

See also

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

References

Further reading

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External links

* Functions to evaluate half-''t'' distributions are available in several R packages, e.g

Continuous distributions

{{stats-stub, date=August 2016