# Generalized Hermite distribution

### Description

Probability mass, distribution and quantile functions; random generation; and regression models for the generalized Hermite distribution.

### Details

The package implements probability mass function dhermite, distribution function phermite, quantile function qhermite and random generation rhermite for the generalized Hermite distribution. The probability mass function is usually parametrized in terms of the mean μ and the index of dispersion d = \frac{σ^2}{μ}:

P(X=x) = P(X=0) \frac{μ^x (m-d)^x}{(m-1)^x} ∑_{j=0}^{[x/m]} \frac{(d-1)^j (m-1)^{(m-1)j}}{m^j μ^{(m-1)j} (m-d)^{mj} (x-mj)!j!} where P(X=0) = exp(μ (-1+ \frac{d-1}{m})), m is the degree of the generalized Poisson distribution and [x/m] is the integer part of x/m.

The package is able to fit Hermite regression models as well, by means of the function glm.hermite, also in the presence of covariates.

### Author(s)

David Moriña, Manuel Higueras, Pedro Puig and María Oliveira

Mantainer: David Moriña Soler <david.morina@uab.cat>

### References

Kemp C D, Kemp A W. Some Properties of the Hermite Distribution. Biometrika 1965;52 (3-4):381–394.

McKendrick A G Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 1926;44:98–130.

Kemp A W, Kemp C D. An alternative derivation of the Hermite distribution. Biometrika 1966;53 (3-4):627–628.

Patel Y C. Even Point Estimation and Moment Estimation in Hermite Distribution. Biometrics 1976;32 (4):865–873.

Gupta R P, Jain G C. A Generalized Hermite distribution and Its Properties. SIAM Journal on Applied Mathematics 1974;27:359–363.

Bekelis, D. Convolutions of the Poisson laws in number theory. In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania 1996;4:283–296.

Zhang J, Huang H. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics 2013;2:110–121.

Kotz S. Encyclopedia of statistical sciences. John Wiley 1982-1989.

Kotz S. Univariate discrete distributions. Norman L. Johnson 2005.

Puig P. (2003). Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, with an Application to Generalized Hermite Distributions. Journal of the American Statistical Association 2003; 98:687–692.

Distributions for some other distributions, qhermite, phermite, rhermite, hermite-package, glm.hermite

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