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Normal-inverse Gaussian distribution

A four-parameter probability law formed by mixing a normal mean and variance with an inverse Gaussian variable.

Version
v1 · 2026-10-31 · History
Domain-specific #
14538
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Probability Theory, Parametric Distributions → Mathematics
Aliases
NIG distribution

Core Idea

A normal-inverse Gaussian (NIG) distribution is a probability law formed by mixing a conditional normal distribution with a positive inverse Gaussian variable. Given \(V\), the outcome \(X\) is normal with mean \(\mu+\beta V\) and variance \(V\); averaging over \(V\) gives the marginal law. The mathematical mixer is not automatically a measured physical cause.[1][2]

The four parameters describe tail shape \(\alpha\), asymmetry \(\beta\), scale \(\delta\) and location \(\mu\), with \(\alpha>|\beta|\) and \(\delta>0\) in the stated parameterization. Setting \(\beta=0\) gives a symmetric member.[3][2]

How would you explain it like I'm…

 

No faithful explanation at this level. All three agree a five-year-old picture must present a real hidden thing that stretches and shifts a bell curve, collapsing the mathematical mixture representation into a physical mixing process.

Blended Bell Curves

A probability distribution describes which results are likely and which are rare. The normal-inverse Gaussian distribution is built with a math recipe: first a positive random number is chosen from one kind of distribution, and then that number sets both the center and the spread of a bell curve. Averaging over all the possible first numbers gives the final shape. Its four settings control how heavy the tails are, how lopsided it is, how wide it is and where it sits. The recipe is just a way to build the shape — it doesn't mean there's a real hidden number being measured.

Normal Mixed by Inverse Gaussian

The normal-inverse Gaussian (NIG) distribution is a continuous probability distribution built by mixing. Start with a positive random variable V from an inverse Gaussian distribution. Given V, let X be normal with mean μ + βV and variance V. Averaging over all values of V gives the distribution of X alone, called its marginal law. It has four parameters: α (tail shape), β (asymmetry), δ (scale) and μ (location), with α > |β| and δ > 0. This is a mathematical construction; it doesn't by itself mean there is a measurable physical mixing variable. The NIG is one special member of a larger family called generalized hyperbolic distributions.

 

The normal-inverse Gaussian distribution is a continuous law obtained as a normal variance-mean mixture: conditional on a positive inverse Gaussian mixing variable V, X is normal with mean μ + βV and variance V, with the inverse Gaussian parameters chosen consistently with the NIG scale and tail parameters; integrating over V yields the marginal law of X. The representation is mathematical and does not by itself identify a measured physical mixing variable. The conventional parameters are α (tail shape), β (asymmetry), δ (scale) and μ (location), subject to α > |β| and δ > 0, with γ = √(α² − β²). The density is f(x) = [αδ K₁(α√(δ² + (x − μ)²)) / (π√(δ² + (x − μ)²))] exp(δγ + β(x − μ)), where K₁ is a modified Bessel function of the second kind. The NIG is the λ = −1/2 member of the generalized hyperbolic family, and the density defines a family of laws independently of any fitted data. A homogeneous NIG Lévy process can be constructed, but other processes with NIG marginals can have different dependence structures, which the one-time law does not determine.

Structural Signature

  • Outcome: a real-valued random variable with an NIG marginal law.
  • Conditional kernel: a normal distribution conditional on a positive variable \(V\).
  • Mixing law: \(V\) has an inverse Gaussian distribution.
  • Coupling: \(V\) controls conditional variance and shifts the mean through \(\beta V\).
  • Marginal: integrating over \(V\) yields a four-parameter law; a fitted parameter vector is one member of the family.[1][2][3]

What It Is Not

A generic heavy-tailed distribution or a normal mixture with another mixer is not automatically NIG. An inverse Gaussian variable alone is the mixer, not the resulting marginal. A time-indexed NIG process adds joint behavior; a fit to one marginal does not establish independent increments or a literal physical mixing mechanism.[1][4]

Scope of Application

The law can be specified mathematically without data, then fitted to real-valued observations. Its use in financial returns or solar-wind increments does not imply that either system contains an observed inverse Gaussian clock.[5][3]

Clarity

Distinguish the family, a member with fixed parameters, a fit to data and a process over time. A symmetric member remains NIG; skewness is possible, not mandatory.[2][3]

Examples

Symmetric member. Choose \(\beta=0\), \(\alpha>0\), \(\delta>0\) and any real \(\mu\). The conditional normal and inverse Gaussian mixture still define an NIG law, now symmetric about its location.[2][3]

Two fitted settings. One study compared NIG with other generalized-hyperbolic subclasses for high-frequency IBEX35 returns. Another fitted NIG densities to solar-wind Elsasser increments at multiple scales. These are case-specific marginal fits, not proof of a common physical mixer or a complete temporal model.[5][3]

Reasoning and Boundary

Check the admissible density or equivalent conditional-normal/inverse-Gaussian representation before calling a law NIG. Then distinguish an estimated marginal from a time-indexed process. A common family across scales does not require one fixed parameter vector.[2][3][4]

Neighborhood in Abstraction Space

Normal-inverse Gaussian distribution sits in a sparse region of the domain-specific corpus (94th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2835 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-31

References

[1] Ole E. Barndorff-Nielsen, “Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling”, Scandinavian Journal of Statistics 24 (1997), 1–13. Publisher abstract and metadata inspected; the full original article was not available in this pass. The publisher lists a 2007 online date for the 1997 volume. registry ↩a ↩b ↩c

[2] SciPy, scipy.stats.norminvgauss documentation, v1.11.3, Notes. Official implementation documentation used as an independent check of the conditional representation and standardized/four-parameter conventions, not as the original discovery source. registry ↩a ↩b ↩c ↩d ↩e ↩f

[3] Juan C. Palacios, Jean C. Perez, and Sofiane Bourouaine, “Parametric description of intermittent probability distribution functions in solar wind and magnetohydrodynamic turbulence”, Monthly Notices of the Royal Astronomical Society 531 (2024), 24–34. Publisher full text §§3–5, especially §4.2 equations (26)–(29), inspected. The source uses Λ where this entry uses δ for scale. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g

[4] Ole E. Barndorff-Nielsen, “Processes of normal inverse Gaussian type”, Finance and Stochastics 2 (1997), 41–68. Publisher abstract and metadata inspected; it discusses multiple NIG-marginal process types with different dependence structures, not a particular empirical financial fit. registry ↩a ↩b

[5] Pablo Suárez-García and David Gómez-Ullate, “Scaling, stability and distribution of the high-frequency returns of the IBEX35 index”, author manuscript, published in Physica A 392 (2013), 1409–1417. Abstract, §2.2, and §3 inspected (one-based PDF pp. 1, 3–6); the comparative conclusion is limited to their data and tested families. registry ↩a ↩b