# Generalized Hermite distribution

### Description

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

### Details

Package: | hermite |

Type: | Package |

Version: | 1.1.0 |

Date: | 2015-03-24 |

License: | GPL version 2 or newer |

LazyLoad: | yes |

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.

### See Also

`Distributions`

for some other distributions,
`qhermite`

, `phermite`

, `rhermite`

,
`hermite-package`

, `glm.hermite`

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