GeneralizedHyperbolic

The Generalized Hyperbolic (GH) distribution is the most general member of the family — the normal variance-mean mixture

\[ X \mid Y \sim \mathcal{N}(\mu + \gamma Y,\; \Sigma Y), \qquad Y \sim \mathrm{GIG}(p, a, b), \]

with a full GIG subordinator. Because the GIG nests the Gamma, InverseGamma, and InverseGaussian as limits, GH contains the VarianceGamma, NormalInverseGamma, and NormalInverseGaussian as special cases — reach for it when you want the most flexible model.

Density gallery

Beyond the shared location \(\mu\), skewness \(\gamma\), and dispersion \(\Sigma\), GH exposes the GIG shape \(p\), which tunes the tail weight independently. Here we vary \(p\) at a fixed positive skew.

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import jax
jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
import numpy as np
import matplotlib.pyplot as plt

from normix import GeneralizedHyperbolic, UnivariateGeneralizedHyperbolic
from normix.utils.plotting import set_theme

set_theme()
np.set_printoptions(precision=5, suppress=True)

x = jnp.linspace(-6.0, 6.0, 400)
fig, ax = plt.subplots()
for p in [-1.5, -0.5, 1.0, 3.0]:
    gh = UnivariateGeneralizedHyperbolic.from_classical(
        mu=0.0, gamma=0.5, sigma=1.0, p=p, a=1.0, b=1.0)
    ax.plot(np.asarray(x), np.asarray(jax.vmap(gh.pdf)(x)), label=f"p = {p:g}")
ax.set_xlabel("x"); ax.set_ylabel("density")
ax.set_title(r"GeneralizedHyperbolic marginals ($\mu$=0, $\gamma$=0.5, $\sigma$=1, $a$=$b$=1)")
ax.legend()
plt.show()

Parametrizations

Built from the shared location/shape block and the GIG subordinator:

Symbol	Attribute	Meaning
\(\mu\)	mu	location \((d,)\)
\(\gamma\)	gamma	skewness \((d,)\)
\(\Sigma = L_\Sigma L_\Sigma^\top\)	L_Sigma	dispersion Cholesky \((d, d)\)
\(p\)	p	GIG shape (any real)
\(a\)	a	GIG rate (\(> 0\))
\(b\)	b	GIG rate (\(> 0\))

The marginal is not an exponential family, but the joint \((X, Y)\) is — its natural parametrization \(\theta\), the closed-form Bessel log-density, and the EM/MCECM algorithms are derived in The Generalized Hyperbolic Distribution and EM Algorithm for Generalized Hyperbolic Distributions.

Quick usage

Raw \((\gamma, \Sigma, a, b)\) are identified only up to the scale gauge \(Y \mapsto cY\) (see EM Algorithm for Generalized Hyperbolic Distributions); compare the invariants \(\mu\), \(\gamma E[Y]\), and \(E[Y]\,\Sigma\) instead.

mu = jnp.array([0.0, 0.0])
gamma = jnp.array([0.3, -0.4])
Sigma = jnp.array([[1.0, 0.3], [0.3, 1.0]])

gh = GeneralizedHyperbolic.from_classical(mu=mu, gamma=gamma, sigma=Sigma, p=-0.5, a=1.0, b=1.0)
print("mean:", np.asarray(gh.mean()))
print("cov:\n", np.asarray(gh.cov()))

X = gh.rvs(2_000, seed=0)
# default_init warm-starts from the best of the NIG / VG / NInvG sub-model fits
result = GeneralizedHyperbolic.default_init(X).fit(X, max_iter=50, tol=1e-3)
fit = result.model
ey = float(fit.joint.subordinator().mean())
print("converged:", bool(result.converged), "n_iter:", int(result.n_iter))
print("mu:", np.asarray(fit.mu))
print("gamma * E[Y]:", np.asarray(fit.gamma) * ey)   # true [0.3, -0.4] (E[Y]=1)
print("E[Y] * Sigma:\n", ey * np.asarray(fit.sigma()))
print("mean:", np.asarray(fit.mean()))
print("cov:\n", np.asarray(fit.cov()))

mean: [ 0.3 -0.4]
cov:
 [[1.09 0.18]
 [0.18 1.16]]
converged: True n_iter: 8
mu: [-0.01877  0.03617]
gamma * E[Y]: [ 0.30823 -0.40721]
E[Y] * Sigma:
 [[0.93183 0.26059]
 [0.26059 0.94615]]
mean: [ 0.28947 -0.37104]
cov:
 [[1.03602 0.12295]
 [0.12295 1.12798]]

The \(d = 1\) sibling UnivariateGeneralizedHyperbolic adds cdf / ppf; the FactorGeneralizedHyperbolic scales to high dimensions.

See also

• API: GeneralizedHyperbolic

• Theory: The Generalized Hyperbolic Distribution, EM Algorithm for Generalized Hyperbolic Distributions

• Tutorials: A tour of the GH family, Normal variance-mean mixtures