Halphen Distribution

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The Halphen distribution, also called the generalized inverse Gaussian distribution, is a right-skewed, heavy-tailed distribution defined only for positive real variables. It approaches 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.

Type B

Type-IB

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.