In statistics, the folded-''t'' and half-''t'' distributions are derived from Student's ''t''-distribution by taking the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

s of variates. This is analogous to the folded-normal and the half-normal

statistical distribution In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution functio ...

s being derived from the

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

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 In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

is given by: :

\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

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 In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

) are given by: :

;=\;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 distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the hal ...

s'' by allowing for finite degrees-of-freedom (the

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

analogues constitute the limiting cases of infinite degrees-of-freedom). Since the

Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...

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

References

External links

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

Continuous distributions {{stats-stub, date=August 2016

Definitions

Relation to other distributions

See also

References

Further reading

External links