Folded-t And Half-t Distributions
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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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
s of variates. This is analogous to the folded-normal and the
half-normal 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 ...
statistical distribution In statistics, an empirical distribution function ( an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step functio ...
s being derived from the
normal distribution In probability theory and 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 ...
.


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), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
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 In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
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 expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
) 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 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 ha ...
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-Louis 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) ...
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 The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal d ...
*
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 ha ...
*
Modified half-normal distribution In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple ...
*
Half-logistic distribution In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. That is, for :X = , Y, \! whe ...


References


Further reading

* * * * *


External links

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

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