Hermite distribution

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The Hermite distribution is a generalization of the Poisson distribution for modeling count data with more than one parameter. It is especially useful for modeling count data that is overdispersed or multimodal — most distributions that deal with count data, such as the Poisson distribution, cannot handle multimodality. The name “Hermite” comes from the fact that the distribution’s probability function and moment generating function can be expressed as coefficients of modified Hermite polynomials [1].

Plot of a Hermite distribution [2].

Hermite distribution properties

A convenient expression for the probability mass function (PMF) is [3]

hermite distribution PMF

Where [r/2] is the integer part of r/2 and ! is a factorial.

The PMF can also be expressed in terms of Hermite polynomials [4]:

where

  • Hn = Hermite polynomials
  • a, b = means
  • X, Y = two independent Poisson variables.

Other properties:

The generalized Hermite distribution (GHD) can be obtained as a Poisson-Binomial distribution with parameter m for the number of data points in the cluster. Alternatively, it can be obtained as the sum of m correlated Poisson random variables (X1 + Z, …, Xm + Z) with probability generating function (PGF)

Where

θ1 ≥ 0, θ2 ≥ 0, m ∈ ℤ+ (θ2 = 0 corresponds to the Poisson distribution). The PMF for the generalized distribution is given by

Generalized PMF; the Hermite corresponds to m = 2 [2].

Relationship to other distributions

The zero-inflated Poisson can handle excess counts of zeros, while the Hermite distribution can handle an excess of counts for several values.

  • When γ = 0, the Hermite PMF becomes the Poisson distribution.
  • The Hermite distribution is a special case of the Poisson Binomial distribution for n = 2.
  • The bivariate Poisson distribution and the Poisson-binomial distribution are special cases of the Hermite distribution [5]. 
  • The Hermite distribution can be obtained by combining a Poisson and a normal distribution [6].
  • The Hermite is also a convolution of an ordinary and a doublet Poisson and the sum of two correlated Poissons [7].

History of the Hermite distribution

The Hermite distribution first appeared in McKendrick’s 1926 paper Applications of Mathematics to Medical Problems, in the context of medical research [8]. McKendrick found that the distribution of the sum of two correlated Poisson variables provided a good fit for leucocyte bacteria counts. The distribution was formally named the Hermite distribution in 1965, in Kemp and Kemp’s 1965 paper Some Properties of ‘Hermite’ Distribution.

References

[1] Hermite distribution image: Estefany87, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

[2] Kemp, C.D; Kemp, A.W (1965). “Some Properties of the “Hermite” Distribution”. Biometrika. 52 (3–4): 381–394. doi:10.1093/biomet/52.3-4.381.

[3] Grzybek, P. & Kohler, R. Exact Methods in the Study of Language and Text. De Gruyter.

[4] Cariolaro, G. 2015. Quantum Communications. Springer International.

[5] Giles, D. 2010. Hermite regression analysis of multi-modal count data. Economics Bulletin, 30, 2936-2945.

[6] Kemp, ADRIENNE W.; Kemp C.D (1966). “An alternative derivation of the Hermite distribution”. Biometrika. 53 (3–4): 627–628. doi:10.1093/biomet/53.3-4.627.

[7] Charalambides, C. 2005. Combinatorial Methods in Discrete Distributions. Wiley.

[8] McKendrick, A.G. (1926). “Applications of Mathematics to Medical Problems”. Proceedings of the Edinburgh Mathematical Society. 44: 98–130. doi:10.1017/s0013091500034428.

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