< List of probability distributions < *Halphen distribution*

The **Halphen distribution**, also called the *generalized inverse Gaussian distribution*, is a right-skewed, heavy-tailed distribution defined only for positive real variables. It tends to zero for small values of *x*.

The invention of the Halphen distribution is attributed to French statistician and hydrologist Étienne Halphen [1] by some authors [e.g., 2, 3]. However, other authors [e.g., 4] attribute its invention to Good [5] via Jorgensen [6], who do not use the “Halphen” moniker.

## Halphen distribution PDF

The Halphen distribution is a family of three distributions: Type-A, Type-B and Type-IB. Their probability density functions (PDFs) are as follows [7]:

**Type A**

Where:

- x > 0
- m = strictly positive scale parameter (strictly positive meaning it must be greater than 0)
- α and ν ∈ ℝ = shape parameters (α is strictly positive)
- K
_{ν}(x) = modified Bessel function of second kind of order ν.

Where:

- x > 0
- m = strictly positive scale parameter
- ν = strictly positive shape parameter
- α = shape parameter α ∈ ℝ
- ef
_{ν}(x) = exponential factorial function *.

*Halphen defined the exponential factorial function as

The gamma distribution is a limiting case between Type-A and Type-B of the Halphen distribution; the inverse Gamma distribution is a limiting case between type-A and type-IB.

## References

[1] Halphen, E. (1941). Sur un nouveau type de courbe de fréquence. Compte-Rendus de l’Académie des Sciences, 213, 633-635.

[2] Morlat, G. (1956). Les lois de probabilites de Halphen, Revue de Statistique Appliquee, Vol 4, No 3, 21-46.

[3] Seshadri, V. (1997). Halphen’s laws. In: Kotz, S. et al. Encyclopedia of Statistical Sciences, Update, Vol 1 302-306. Wiley.

[4] Chaudry, M. & Zubair, S. (1992). Two Integrals Arising in Generalized Inverse Gaussian Model and Heat Conduction Problems. Vol 34. Issue 3.

[5] Good, I. (1953). The population frequencies of species and the elimination of population parameters. Biometrika, Vol. 40, 237-260.

[6] Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics, Vol. 9. Springer.

[7] Delhome, R. et al. (2017). Travel time statistical modeling with the Halphen distribution family. Journal of Intelligent Transportation Systems Technology Planning and Operations · May.