Maximum likelihood estimation and Hermite regression


glm.hermite is used to fit generalized linear models with count responses following a Hermite distribution, specified by giving a symbolic description of the linear predictor. A summary method providing the most meaningful information on the fitted model is available for objects of class glm.hermite.


  glm.hermite(formula, data, link="log", start=NULL, m = NULL)



symbolic description of the model. A typical predictor has the form response ~ terms where response is the (numeric) response vector and terms is a series of terms which specifies a linear predictor for response.


an optional data frame containing the variables in the model.


character specification of link function: "log" or "identity". By default link="log".


a vector containing the starting values for the parameters of the specified model. Its default value is NULL.


value for parameter m. Its default value is NULL, and in that case it will be estimated inside the function.


glm.hermite returns an object of class glm.hermite, which is a list including the following components:

  • coefs the vector of coefficients.

  • data an optional data frame containing the variables in the model.

  • loglik log-likelihood of the fitted model.

  • vcov covariance matrix of all coefficients in the model (derived from the Hessian of the maxLik output).

  • hessHessian matrix, returned by the maxLik output.

  • fitted.values the fitted mean values, obtained by transforming the linear predictors by the inverse of the link function.

  • wLikelihood ratio test statistic.

  • pvalLikelihood ratio test p-value.


María Oliveira, Manuel Higueras, David Moriña and Pere Puig


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


data <- c(rep(0,122), rep(1,40), rep(2,14), rep(3,16), rep(4,6), rep(5,2))
mle1 <- glm.hermite(data~1, link="log", start=NULL, m=3)
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