`scipy.stats._continuous_distns:genhyperbolic_gen`

source: /scipy/stats/_continuous_distns.py :3982

Signature

   class     genhyperbolic_gen (    momtype    =  1   ,    a    =  None   ,    b    =  None   ,    xtol    =  1e-14   ,    badvalue    =  None   ,    name    =  None   ,    longname    =  None   ,    shapes    =  None   ,    seed    =  None     )

Members

Summary

A generalized hyperbolic continuous random variable.

Extended Summary

%(before_notes)s

Notes

The probability density function for genhyperbolic is:

f (x, p, a, b) = \frac{( a ^{2} - b ^{2} ) ^{p /2}}{2 π a ^{p - 1/2} K _{p} ( a ^{2} - b ^{2} )} e^{b x} \times \frac{K _{p - 1/2} ( a 1 + x ^{2} )}{( 1 + x ^{2} ) ^{1/2 - p}}

for $x, p \in (- \infty; \infty)$ , $∣ b ∣ < a$ if $p \geq 0$ , $∣ b ∣ \leq a$ if $p < 0$ . $K_{p} (.)$ denotes the modified Bessel function of the second kind and order $p$ (scipy.special.kv)

genhyperbolic takes p as a tail parameter, a as a shape parameter, b as a skewness parameter.

%(after_notes)s

The original parameterization of the Generalized Hyperbolic Distribution is found in ^[1] as follows

f (x, λ, α, β, δ, μ) = \frac{( γ / δ ) ^{λ}}{2 π K _{λ} ( δ γ )} e^{β (x - μ)} \times \frac{K _{λ - 1/2} ( α δ ^{2} + ( x - μ ) ^{2} )}{( δ ^{2} + ( x - μ ) ^{2} / α ) ^{1/2 - λ}}

for $x \in (- \infty; \infty)$ , $γ := α^{2} - β^{2}$ , $λ, μ \in (- \infty; \infty)$ , $δ \geq 0, ∣ β ∣ < α$ if $λ \geq 0$ , $δ > 0, ∣ β ∣ \leq α$ if $λ < 0$ .

The location-scale-based parameterization implemented in SciPy is based on ^[2], where $a = α δ$ , $b = β δ$ , $p = λ$ , $sc a l e = δ$ and $l oc = μ$

Moments are implemented based on ^[3] and ^[4].

For the distributions that are a special case such as Student's t, it is not recommended to rely on the implementation of genhyperbolic. To avoid potential numerical problems and for performance reasons, the methods of the specific distributions should be used.

Aliases

scipy.stats._continuous_distns.genhyperbolic_gen