NormalInverseGaussian

The Normal-Inverse Gaussian (NIG) distribution is the normal variance-mean mixture

\[ X \mid Y \sim \mathcal{N}(\mu + \gamma Y,\; \Sigma Y), \qquad Y \sim \mathrm{InvGaussian}(\mu_{IG}, \lambda), \]

with an InverseGaussian subordinator — the GIG case \(p = -\tfrac12,\; a = \lambda/\mu_{IG}^2,\; b = \lambda\). It combines semi-heavy tails with tractable moments and is the standard robust default for financial returns.

Density gallery

The skewness \(\gamma\) tilts the marginal; the InverseGaussian mixing gives it its characteristic semi-heavy tails.

Show code cell source

Hide code cell source

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 NormalInverseGaussian, UnivariateNormalInverseGaussian
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 g in [-1.5, 0.0, 1.5]:
    nig = UnivariateNormalInverseGaussian.from_classical(
        mu=0.0, gamma=g, sigma=1.0, mu_ig=1.0, lam=1.5)
    ax.plot(np.asarray(x), np.asarray(jax.vmap(nig.pdf)(x)), label=rf"$\gamma$ = {g:g}")
ax.set_xlabel("x"); ax.set_ylabel("density")
ax.set_title(r"NormalInverseGaussian marginals ($\mu$=0, $\sigma$=1, $\mu_{IG}$=1, $\lambda$=1.5)")
ax.legend()
plt.show()

Parametrizations

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

Symbol	Attribute	Meaning
\(\mu\)	mu	location \((d,)\)
\(\gamma\)	gamma	skewness \((d,)\)
\(\Sigma = L_\Sigma L_\Sigma^\top\)	L_Sigma	dispersion Cholesky \((d, d)\)
\(\mu_{IG}\)	mu_ig	InverseGaussian mean
\(\lambda\)	lam	InverseGaussian shape

The marginal is not an exponential family, but the joint \((X, Y)\) is — its natural parametrization and the EM machinery are in The Generalized Hyperbolic Distribution.

Quick usage

Raw \((\gamma, \Sigma, \mu_{IG})\) 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]])

nig = NormalInverseGaussian.from_classical(mu=mu, gamma=gamma, sigma=Sigma, mu_ig=1.0, lam=1.5)
print("mean:", np.asarray(nig.mean()))
print("cov:\n", np.asarray(nig.cov()))
print("skewness:", np.asarray(nig.skewness()))
print("excess kurtosis:", np.asarray(nig.kurtosis()))

X = nig.rvs(2_000, seed=0)
result = NormalInverseGaussian.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:
 [[1.06    0.22   ]
 [0.22    1.10667]]
skewness: [ 0.58277 -0.76047]
excess kurtosis: [2.45283 2.77108]
converged: True n_iter: 18
mu: [0.03145 0.00892]
gamma * E[Y]: [ 0.2587  -0.39766]
E[Y] * Sigma:
 [[1.01448 0.28978]
 [0.28978 1.03098]]
mean: [ 0.29015 -0.38873]
cov:
 [[1.07348 0.19909]
 [0.19909 1.17039]]

The \(d = 1\) sibling UnivariateNormalInverseGaussian adds cdf / ppf for VaR-style tail calculations; the FactorNormalInverseGaussian handles many assets.

See also

• API: NormalInverseGaussian

• Theory: The Generalized Hyperbolic Distribution

• Tutorials: Normal variance-mean mixtures, A heavy-tailed index series