Maximum likelihood estimation and Hermite regression

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Description

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.

Usage

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

Arguments

formula

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.

data

an optional data frame containing the variables in the model.

link

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

start

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

m

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

Value

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.

Author(s)

María Oliveira, Manuel Higueras, David Moriña and Pere Puig

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

Examples

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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)
mle1