`scipy.stats._continuous_distns:norminvgauss_gen`

source: /scipy/stats/_continuous_distns.py :5468

Signature

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

Members

Summary

A Normal Inverse Gaussian continuous random variable.

Extended Summary

%(before_notes)s

Notes

The probability density function for norminvgauss is:

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

where $x$ is a real number, the parameter $a$ is the tail heaviness and $b$ is the asymmetry parameter satisfying $a > 0$ and $∣ b ∣ <= a$ . $K_{1}$ is the modified Bessel function of second kind (scipy.special.k1).

%(after_notes)s

A normal inverse Gaussian random variable Y with parameters a and b can be expressed as a normal mean-variance mixture: Y = b * V + sqrt(V) * X where X is norm(0,1) and V is invgauss(mu=1/sqrt(a**2 - b**2)). This representation is used to generate random variates.

Another common parametrization of the distribution (see Equation 2.1 in ^[2]) is given by the following expression of the pdf:

g (x, α, β, δ, μ) = \frac{α δ K _{1} ( α δ ^{2} + ( x - μ ) ^{2} )}{π δ ^{2} + ( x - μ ) ^{2}} e^{δ α^{2} - β^{2} + β (x - μ)}

In SciPy, this corresponds to $a = α δ, b = β δ, loc = μ, scale = δ$ .

Aliases

scipy.stats._continuous_distns.norminvgauss_gen