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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 values of variates. This is analogous to the folded-normal and the half-normal statistical distributions being derived from the normal distribution.

Contents

Definition

The folded non-standardized t distribution is the distribution of the absolute value of the non-standardized t distribution with ν degrees of freedom; its probability density function is given by:

g ( x ) = Γ ( ν + 1 2 ) Γ ( ν 2 ) ν π σ 2 { [ 1 + 1 ν ( x − μ ) 2 σ 2 ] − ν + 1 2 + [ 1 + 1 ν ( x + μ ) 2 σ 2 ] − ν + 1 2 } ( for x ≥ 0 ) .

The half-t distribution results as the special case of μ = 0 , and the standardized version as the special case of σ = 1 .

Relation to normal and Cauchy 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 and half-Cauchy distributions for ν = 1 .

The half-t distribution

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

g ( x ) = 2 Γ ( ν + 1 2 ) Γ ( ν 2 ) ν π σ 2 ( 1 + 1 ν x 2 σ 2 ) − ν + 1 2 ( for x ≥ 0 ) .

The half-t distribution's first two moments (expectation and variance) are given by:

E ⁡ [ X ] = 2 σ ν π Γ ( ν + 1 2 ) Γ ( ν 2 ) ( ν − 1 ) for ν > 1 ,

and

Var ⁡ ( X ) = σ 2 ν ν − 2 4 ν π ( ν − 1 ) 2 ( Γ ( ν + 1 2 ) Γ ( ν 2 ) ) 2 for ν > 2 .

References

Folded-t and half-t distributions Wikipedia
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